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Splitting and time reversal

# Splitting and time reversal for Markov additive processes

## Abstract.

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into behaviour of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener-Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Assuming no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line using three different approaches, which further demonstrates applicability of path decomposition theory.

###### Key words and phrases:
Lévy process, path decomposition, time reversal, Wiener-Hopf factorization, last exit, conditioned to stay positive
Financial support by the Swiss National Science Foundation Project 200020 143889 is gratefully acknowledged.

## 1. Introduction

In the theory of Markov processes path decomposition or splitting time theorems have a long history, see e.g. [21, 25] and references therein. For a nice Markov process these results are concerned with a study of random times such that the post- process has a Markovian structure and is independent of the events before given (and sometimes also ). The main examples of such are stopping times, the time of the infimum and the last exit time from , all of which fall in the category of so-called randomized coterminal times [21]. The first example is rather trivial, and the latter two have the same structure, i.e. the time of the infimum can be seen as the last exit time from a random interval . Thus it is not surprising that they lead to the same law for the post- process: the law of conditioned to stay above a certain level.

For a Lévy process the theory of path decomposition becomes especially compelling and rich with various relations, see [22, 10] and also [4, Ch. VII.4], for the one-sided case. It provides a valuable insight in the behaviour of the process and proves to be useful in various computations. In particular, the celebrated Wiener-Hopf factorization is just an application of splitting at the infimum, see [22, 11] or [4, Lem. VI.6]. The aim of this work is to develop the corresponding theory for Markov additive processes (MAPs), which are the processes with stationary and independent increments given the current phase. These processes often appear in finance, queueing and risk theories [2], and recently were found to be instrumental in the analysis of real self-similar Markov processes, see  [8]. Even though MAPs closely resemble Lévy processes, their increments are not exchangeable, and hence one may expect certain difficulties as well as apriory non-obvious differences from the Lévy case.

### 1.1. Overview of the results

The results of this paper are formulated for defective MAPs, where general phase-dependent killing is allowed. This is important in applications since it permits tracking time spent in different phases by way of joint Laplace transforms, see Section 2.1. In the following we provide a brief overview of the main results and point out the main differences/difficulties as compared to the case of a Lévy process, see also Figure 1.

• Splitting and conditioning:

• Section 3.1 presents splitting at the infimum and defines the law of the post-infimum process given phase . There may be a phase switch at the infimum, see Figure 2, and it is crucial to condition on the correct phase. Moreover, it is convenient to choose this phase in a slightly different way for infimum and supremum.

• Section 3.2 shows that corresponds to the original process conditioned to stay positive. For certain phases this process starts from a positive level and possibly different phase. This initial distribution is addressed in Proposition 3.3.

• Section 3.3 shows that splitting also holds at the last exit time from an interval given the exit is continuous, and then the law of the post- process is . Thus post-infimum and post- processes have the same law given that they start in the same phase  (continuous exit can be realized only in some phases).

• Time reversal and equivalence of laws:

• Section 4.1 discusses time reversal at the life time of the process. In general, the life time depends on the process and hence the time reversal identity is different from the classical formula concerning reversal at an independent time.

• Section 4.2 expresses the law of the process reversed at the infimum via the dual process conditioned to stay non-positive, see Theorem 4.1. Importantly, there are new non-trivial constants in this identity.

• Section 4.3 shows that the process reversed at the last exit is closely related to the dual process considered up to the last exit (on the events of continuous exit), see Theorem 4.2. There is another set of constants which have a clear probabilistic interpretation.

• Section 4.4 completes the list of reversal identities by showing that reversal at the first passage (when continuous) results in the dual process conditioned to stay positive and considered up to the last exit, see Theorem 4.3. Again there is a need for appropriate scaling.

• Wiener-Hopf factorization: The statement of the factorization and its relation to the results in [17, 18, 8] can be found in Section 5. These works consider MAPs killed at independent exponential times, which would be natural for Lévy processes, but is not fully satisfactory for MAPs. Moreover, we allow for counting time in each phase separately. Finally, we provide some necessary corrections to the factorizations in the literature.

• Spectrally negative MAPs: Section 6 further develops the above results in the important case when the process does not have positive jumps. In particular, the Wiener-Hopf factors are given by compact explicit formulas, which are then compared to the results in [9, 20, 13].

• Application: Section 6.3 studies the times spent in different phases up to the last exit  from the negative half-line (together with the phase at ). Our theory allows for three different approaches: (i) is based on splitting at  and law equivalence of post- and post-infimum processes, (ii) is based on the formula for time reversal at , and (iii) is based on splitting at the infimum and the relation between the post-infimum process considered up to its last exit and the dual process reversed at its first passage, see Theorem 4.3.

The above list shows that splitting and time reversal results for MAPs closely resemble analogous results for Lévy processes, which should help in understanding the present theory and its application. Importantly, there are non-trivial differences in each of these results, and we made an attempt to clearly stress these points. Some parts of the proofs are rather standard and so we keep them very brief. Many results, however, cannot be obtained by a simple generalization of the arguments in the Lévy case. This is especially true when it comes to time reversal, see e.g. Proposition 4.1. It should also be noted that Nagasawa’s reversal theory for Markov processes [23] does not provide easy to use tools in our case, and instead we employ probabilistic arguments.

The subject of this work is rather technical, and our main goal is to present the results in a clear and comprehensible way while treating the details with care. Thus most of the results are supplemented with comments and remarks. Moreover, it is often convenient to draw a picture such as Figure 1.

This picture illustrates splitting at the infimum, and the last exit from in a continuous way. Both post-splitting processes lead to the same law corresponding to the process conditioned to stay positive, see grey axes. Time reversal at these times is depicted by small black axes - consider turning the paper upside down.

## 2. Preliminaries and notation

Consider a Markov additive process , where is the additive component taking values in and is the phase taking values in a finite set . The defining property of a MAP reads: for all given the process is independent of and has the same law as the original process given . Moreover, we allow for killed processes by adding an absorbing state , and write to denote the life time of the process. We require the above defining property to hold, but we do not add to . Observe that is a (possibly transient) Markov chain on and assume that it is irreducible. Finally, any MAP can be seen as a Markov-modulated Lévy process: the additive component evolves as some Lévy process while and has a jump distributed as some at the phase switch from to , where all these components and are independent. We refer to [2] and [13] for basic facts about MAPs.

It is common to write for an matrix with elements . It is well known that there exists a matrix-valued function such that for all and at least purely imaginary . It is noted that is the analogue of the Laplace exponent of a Lévy process, and that it characterizes the law of the corresponding MAP. Finally, we define first passage times:

 τx:={t≥0:Xt>x}, τ−x:={t>0:Xt

### 2.1. Defective processes and time spent in different phases

Note that is the transition rate matrix of , and hence is a vector with transition rates into which are called the killing rates. Throughout this work we assume that our MAP is killed/defective:

 Assumption: q≠0, i.e. ζ<∞%a.s.

Note that has a phase-type distribution (one under each ) with phase generator and exit vector , see e.g. [2, Ch. III.4]. Clearly, the life time  depends on the process unless for all , in which case it is independent and exponentially distributed. On top of the implicit killing with rates we sometimes need additional killing with rates ; we write , and so on, meaning that the underlying killing rates are . Finally, some results of this paper (but not all) can be generalized to non-defective processes by taking for all .

For any random time let be the time spent in phase  up to , and write for the corresponding vector. Note that by the properties of an exponential random variable we have

 E[eαXt−⟨β,t⟩;Jt]=Eβ[eαXt;Jt]=eΨβ(α)t.

Moreover, the joint transform of and at the killing time together with times spent in every phase is given by the following expression.

 (1) E[eαXζ−−⟨β,ζ⟩;Jζ−]=∫∞0E[eαXt−⟨β,t⟩;Jt]Δqdt=∫∞0eΨβ(α)tdtΔq =−(Ψβ(α))−1Δq,

where denotes the diagonal matrix with on the diagonal. Here we used the fact that the set of jump points of has 0 Lebesgue measure, and that all the eigenvalues of have negative real parts for . It is noted that (1) should be compared to in the Lévy case, where is an independent exponential random variable with rate , see the left side of (6.17) in [19].

### 2.2. Partition of phases

Recall that for a Lévy process the point 0 is said to be regular for an open or closed set if hits immediately ( event), and is irregular otherwise. The conditions for regularity can be found in [19, Thm. 6.5]. In the following it will be convenient to partition the phases in three groups :

• contains such that is regular for and irregular for under ; the prime example: has bounded variation and positive drift.

• contains such that is irregular for and regular for under ; the prime example: has bounded variation and negative drift, also it can be a compound Poisson process.

• contains such that is regular for and for under ; the prime example: has unbounded variation.

It is noted that if has bounded variation and 0 drift then may belong to any of the three groups.

## 3. Splitting and conditioning

### 3.1. Splitting at extrema

Define the overall infimum and its (last) time, and the overall supremum and its (first) time:

 X––=inft∈[0,ζ){Xt}, G––=sup{t∈[0,ζ):Xt∧Xt−=X––}, ¯¯¯¯¯X=supt∈[0,ζ){Xt}, ¯¯¯¯G=inf{t∈[0,ζ):Xt∨Xt−=¯¯¯¯¯X}.

Note that the distinction between first and last extrema is only necessary if for some the underlying Lévy process is a compound Poisson process; in this case one may similarly consider first infimum and last supremum times.

#### Splitting at the infimum

###### Proposition 3.1.

The following splitting result holds true.

• On the event the process splits at : given the processes and are independent.

• On the event the process splits at : given the processes and are independent.

In the case of a Lévy process the event has probability either 1 or 0, where the first corresponds to processes of types and and the second to type . Hence the splitting result can be formulated for these two cases separately, see [4, Lem. VI.6]. This issue is more complicated for MAPs. Letting

 (2) J––=JG––\rm 1{XG––=X––}+JG––−\rm 1{XG––>X––},

be the phase in which the infimum was achieved, we have the following important observation.

###### Lemma 3.1.

The following dichotomy holds with probability one:

• If then and ,

• If then and .

###### Proof.

Consider separately the cases when the phase is switched and not switched at , see also Remark 3.1. Note also that does not imply that according to the first scenario in Remark 3.1. ∎

It is clear from Lemma 3.1 and Proposition 3.1 that determines the law of the post-infimum process, and there is no need to parameterize it by the phase ; we denote this law by .

###### Definition 3.1.

Let under be distributed as under given .

If for some and all we have then is left undefined. Now Proposition 3.1 states that given the process splits at or at according to and , and the post-infimum process has the law .

###### Remark 3.1.

It is important to understand the cases when may switch at , say from phase to phase . For this to occur must be able to achieve its infimum at an exponential time or must be able to achieve its infimum at 0. According to [22, Prop. 2.1] there are three (non-exclusive) cases:

• and for some ;

• and for some ;

• and for some ,

see Figure 2. Note that in the last scenario one may alternatively split at ; consider the stopping times of phase switches. Finally, the time of killing can be interpreted as the time of phase switch from some to , and if then necessarily .

###### Proof of Proposition 3.1.

Splitting at the infimum for MAPs can be proven along the lines of [22] or [11], and so we keep the proof rather brief.

For consider the epochs when is at the running minimum and is in . These epochs are stopping times, and hence we may apply classical splitting at each . Thus for any bounded functionals we have

 Ei(f{(Xt,Jt),t∈[0,G––]}g{(XG––+t−X––,JG––+t),t≥0};J––=j) =∑nEi(f{(Xt,Jt),t∈[0,Tn]},Tn<∞) ×Ei(g{(XTn+t−XTn,JTn+t),t≥0};XTn+t−XTn>0,t>0) =∑nEi(f{(Xt,Jt),t∈[0,Tn]};Tn<∞,XTn+t−XTn>0,t>0) ×Ej(g{(Xt,Jt),t≥0}|Xt>0,t>0) =Ei(f{(Xt,Jt),t∈[0,G––]};J––=j)Ej(g{(Xt,Jt),t≥0}|Xt>0,t>0),

which establishes splitting at .

Next, consider the case which implies that there is no phase switch at , and that there is no jump of at . In this case we approximate by an array of stopping times. The standard way is to use the inverse local time at the infimum, see e.g. [4, Lem. VI.6] for the Lévy case and [18, 8] discussing the same procedure for MAPs. More precisely, let be a continuous local time process of at . On the event we apply classical splitting at the stopping time similarly to the previous paragraph. Summing up over and taking yields splitting at .

For one may use the same argument as in the previous paragraph. Nevertheless, we present an alternative approach providing a better insight into the post-infimum process. Consider the epochs of jumps of larger than which constitute a sequence of stopping times. Note that a given epoch coincides with if and . But for each of these epochs we can apply classical splitting right-before the jump, since the jump is also independent of the past given the corresponding phase. These ideas result in the following identity

 Ei(f{(Xt,Jt),t∈[0,G––)}g{(XG––+t−X––,JG––+t),t≥0};XG––−XG––−>ϵ,JG––−=j) =Ei(f{(Xt,Jt),t∈[0,G––)};XG––−XG––−>ϵ,JG––−=j) ×Ej(g{(X(ϵ)t,J(ϵ)t),t≥0}|X––(ϵ)>0),

where has the law of the original process started from an independent and this latter pair under denotes the first jump larger than while in phase , and the phase right-after this jump; see Section 3.2.1 for further comments about the distribution of . Finally, let to deduce splitting at . ∎

#### Splitting at the supremum

The result about splitting at the supremum can be obtained by taking process, and adapting the statements and the proofs slightly, where it concerns a compound Poisson process . Note also that when discussing splitting at supremum we assume that the absorbing state is . Define

 (3) ¯¯¯¯J:=\rm 1{X¯¯¯G−=¯¯¯¯X}J¯¯¯¯G−+\rm 1{X¯¯¯G−<¯¯¯¯X}J¯¯¯¯G,

where we take in the indicators as compared to in the definition (2) of ; by convention . This choice turns out to be more convenient when it comes to time reversal at the infimum, see Section 4.2. In fact, the only difference appears when there is a phase switch at but no jump of , see the third scenario of Remark 3.1: now we take the phase right before the switch and not right after.

Next, we note that on the set of probability one it holds that

• if then and ,

• if then and .

Finally, given the process splits at or at according to and , and the law of the post-supremum process is denoted by .

### 3.2. Process conditioned to stay positive

For define as the law of conditioned on and for all . Clearly, under this law is a time-homogeneous Markov process with the semigroup

 (4) P↑x,i(Xt∈dy,Jt=j) =Px,i(Xt∈dy,Jt=j|τ−0=∞) =Px,i(Xt∈dy,Jt=j,t<τ−0)Py,j(τ−0=∞)Px,i(τ−0=∞)

for . Recall that we only consider killed processes, and hence .

###### Remark 3.2.

There is a large body of literature devoted to Lévy processes conditioned to stay positive, where the original process drifts to . This case is more complex than ours and may lead to different laws depending on the way conditioning (on the event of probability 0) is implemented, see [12].

The following result is a straightforward adaptation of [22, Prop. 4.1]. It explains the notation chosen for the law of the post-infimum process. The corresponding result for Lévy processes can also be found in [3, 5].

###### Proposition 3.2.

The post-infimum process, i.e.  under , is a Markov process with transition semigroup defined in (4) for .

It is noted that if then the post-infimum process starts at 0 and leaves 0 immediately, and hence the semigroup determines the law . Moreover, it can be shown that converges on the Skorokhod space of paths to as , see [5, Thm. 2] for a similar statement concerning Lévy processes. This result follows from splitting at the infimum and the fact that which is obtained from the corresponding theory for Lévy processes. Note also that for the semigroup (4) can be extended to include , because in this case Then

 P↑i(⋅)=Pi(⋅|τ−0=∞)=Pi(⋅|X––≥0),

which also follows from the proof of Proposition 3.1.

Finally, if then the corresponding result is less neat. Compared to the Lévy case there is an additional problem that does not necessarily converge to  (think of a model with ). Instead we use an alternative approach described in the following Section.

#### On the initial distribution

Let us focus on the law for , which corresponds to the post-infimum process starting from a positive level at time 0. The proof of Proposition 3.4 implies that

 E↑ig{(Xt,Jt),t≥0}=limϵ↓0Ei(g{(X(ϵ)t,J(ϵ)t),t≥0}|X––(ϵ)>0),

where has the law of the original process started from an independent . Notice that is determined by competing independent exponential clocks:

• for the rate is and the level law is ,

• for the rate is and the level law is ,

• for the rate is and the level equals .

It is convenient to define

 (5) Uij(dx):=\rm 1{i=j}νi(dx)+\rm 1% {i≠j}Ψij(0)P(Uij∈dx),

which we call the jump measure associated to a MAP. Now the above observations lead to the following result.

###### Proposition 3.3.

For it holds that

 P↑i(X0∈dx;J0=j) =1ci+qivj(x)Uij(dx), P↑i(J0=†) =qici+qi,

where

 vj(x):=Pj(X––>−x), ci:=∫∞0Ui⋅(dy)v(y)∈[0,∞).
###### Proof.

Observe that

 P↑i(X0>x;J0=j)=limϵ↓0Pi(X(ϵ)0>x;X––(ϵ)>0,J(ϵ)0=j)/Pi(X––(ϵ)>0) =limϵ↓0Pi(X(ϵ)0>x;vj(X(ϵ)0),J(ϵ)0=j)/(∑jPi(vj(X(ϵ)0);J(ϵ)0=j)+Pi(J(ϵ)0=†)),

where and for , and is the total rate of the corresponding jumps. Hence we have

 (6) P↑i(X0>x;J0=j)=∫∞xvj(y)Uij(dy)/(∫∞0Ui⋅(dy)v(y)+qi),

where the first integral is clearly finite. If then it must be that , but the probabilistic reasoning shows that this is only possible when , i.e. no jumps up in phase  are possible, showing that in fact . ∎

Let us mention that we elaborate on this results a bit further in Section 6.4 in the case of a spectrally positive processes.

#### Process conditioned to stay non-positive

Similarly, the law can be seen as the law of the process conditioned to stay non-positive. There are two features of which are different from those of . Firstly, under the process starts from the level 0 if , but it may still do so for , see (3) and think about the exceptional scenario. Secondly, if is a compound Poisson process then under the process stays at the level 0 before jumping down.

### 3.3. Splitting at the last exit from an interval

Define the last exit time from  by

 σa:=sup{t≥0:Xt≤a}

and note that there are two trivial cases: if for all , and if .

The following result is well-known in the theory of Markov processes, see [21] and references therein.

###### Proposition 3.4.

Conditionally on the process is a Markov process with semigroup which is independent of

This result can be proven in a way similar to the proof of Proposition 3.1, that is, we make use of classical splitting at certain stopping times.

The main problem in relating the laws of the post-infimum and the post- processes lies in the initial distribution of the latter. This problem is avoided if , i.e. the post- process starts at 0.

###### Corollary 3.1.

On the event the process has the law , assuming that this event has positive probability, and in particular .

###### Proof.

It is easy to see that , because otherwise is either a compound Poisson process or a process which visits a level at discrete times (if ever) and goes immediately below this level. In any case the event of interest has 0 probability. Now right-continuity of paths and convergence of towards as implies the result. ∎

Note that the results of Proposition 3.4 and Corollary 3.1 also hold for the process under with the same law for the post- process, see also [4, Cor. VII.19] for the case of Lévy processes with no positive jumps.

Finally, we identify the cases when has positive probability. It is said that a Lévy process admits continuous passage upward (creeps upward) if for some (and then for all), which is equivalent to , assuming that is not a compound Poisson process. Sufficient and necessary conditions for this are given in [19, Thm. 7.11].

###### Lemma 3.2.

Suppose is not monotone and is not a compound Poisson process. Then if and only if admits continuous passage upward. Moreover, on this event there is no jump and no phase switch at .

###### Proof.

Note that must be diffuse at each phase switch, and hence it can not be at the level at these times. Moreover, can not jump onto a level . This proves the second statement. Now the first statement follows from a similar result for Lévy processes, see [22, Prop. 5.1] based on time reversal argument.

Note that we exclude the case when , because one may take to be a process of bounded variation, zero drift and such that . Such a process does not admit continuous passage upward, but the above probability would be positive. A similar problem may arise for if is a compound Poisson process whose jump measure has atoms. ∎

## 4. Time reversal

Throughout this section it will be convenient to work with the canonical probability space, where the sample space is a set of càdlàg paths equipped with the Skorokhod’s topology and . Let us define the killing operator and the reversal operator . Namely,

 kT(ω)t :=\rm 1{t

where and depend on . If then we agree that produces a path identically equal to . Note that inverts both time and space, and that the additive component is reversed in the ‘additive’ sense, see Figure 1. In addition, it leads to a càdlàg path which may start at some level if there is a jump of at ; sometimes we will consider in order to ignore this jump, i.e. when reversing at the life time . Next, for an arbitrary non-negative measurable functional we define

 FT:=F∘kT, ^FT:=F∘rT,

i.e we apply to the original process considered up to time  and to the process reversed at . Finally, note that in general a ’MAP killed at ’ is not a (defective) MAP unless has a very particular law, see Section 2.1.

Assume for a moment that and let be the stationary distribution of . Consider now a deterministic time  and note that neither nor jump at a.s. It is well-known (and easy to see) that if is in stationarity then the process time reversed at is also a MAP (with some law ) killed at :

 E(^Ft,J0=i,Jt=j)=^E(Ft,Jt=i,J0=j).

Therefore, we have the following basic time reversal identity

 (7) πiEi(^Ft,Jt=j)=πj^Ej(Ft,Jt=i),

because is in stationarity under both and . It is easy to see that this identity still holds true if the same is used to kill the process under and ; notice that killing amounts to multiplying functionals and by which clearly preserves the correspondence, because is the vector of total times spent in different phases and so it is invariant under time reversal. Thus we again assume implicit killing with rates in the following, and determine by .

In matrix notation we have

 E[^Ft;Jt]=Δ−1π^E[Ft;Jt]TΔπ.

Taking and noting that we get

 eΨ(α)t=E[eαXt;Jt]=Δ−1π^E[eαXt;Jt]TΔπ=Δ−1πe^Ψ(α)TtΔπ,

which implies a well-known relation

 ^Ψ(α)=Δ−1πΨ(α)TΔπ.

In what follows we develop identities similar to (7) for time reversal at the life time , at the infimum , at the last exit time and at the first passage time assuming the exit and passage are continuous. It must be noted that there is a well-developed theory of time reversal for Markov processes, see [23, 26] and also [24] for an illustration. This theory of Nagasawa builds on potential theory, and thus it requires conversions between potential densities (resolvents) and the corresponding Markov processes, which makes it hard to apply in our case even when a resulting Markov process can be guessed by some other means. Instead, we follow a direct probabilistic path.

### 4.1. Time reversal at the life time

Importantly, one can also time revert the process at the killing time even though depends on the process. In this case the relation becomes slightly different:

 (8) πiqiEi(^Fζ−;Jζ−=j)=πjqj^Ej(Fζ−;Jζ−=i),

which readily follows by integrating over life-time and noting that

 πiqi∫∞0Ei(^Ft;Jt=j)qjdt=πjqj∫∞0^Ej(Ft;Jt=i)qidt

according to (7). Note that , and above we chose the first one for the reason of symmetry only. Note also, that killing at an independent exponential time means that and hence (8) simplifies to (7) with , which is not true for general killing.

### 4.2. Time reversal at the infimum

Similarly to the case of Lévy processes we may express the law of a MAP time reversed at its infimum and supremum through the law of this MAP conditioned to stay non-positive and positive, respectively. These identities follow from time reversal at the killing time together with splitting at the extrema. Importantly, there is a new term in these identities as compared to the Lévy case, see the definition of below.

Recall that splitting at the infimum holds either at or depending on the scenario, see also Figure 2. This difficulty is resolved using the following definition:

 ^F–– :=^FG––\rm 1{XG––=X––}+^FG––−\rm 1{XG––>X––},

which has to be compared with the definition of .

###### Theorem 4.1.

It holds that

 πiqiEi(^F––;J––=j)=cj^E↓j(Fζ−;Jζ−=i),

where .

###### Proof.

We may assume that the entries of are strictly positive, because the other case follows by taking limits. Time reversal at the killing time (8) and splitting at the infimum and also at the supremum for the reversed process yield

 πiqiEi(^F––;J––=j)P↑j(Jζ−=k)=πkqk^Pk(¯¯¯¯J=j)^E↓j(Fζ−;Jζ−=i).

Here we have used the fact that coincides with , see the corresponding definitions (2) and (3) and check the third scenario of Figure 2. In particular, we also have

 πiqiPi(J––=j)^P↓j(Jζ−


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# MongoDB and Mathematica

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Posted 6 years ago
 Hi Steeve,I am not aware of a direct link between mathematica and MongoDB but could I recommend using JLink and then writting your mongo interface through Java?Even though it is not a complete solution I know MongoDB does have a REST interface that you could interface through... however it has no support for insert/update/remove operations... But you could read your data at least and you can use Mathematica's ability to communicate through HTTP requests to read the data.If you aren't completely locked into Mongo you could also look at couchdb (http://couchdb.apache.org/) which does have a fully formed REST API. It also takes an optimistic approach to data and instead of locking an entire local instance of the database to write data it versions data which also allows for ACID compliance as opposed to Mongo's partial ACID compliance. All of this being said CouchDB does seem to only allow you to query through map/reduce (and often you just have to write the map function. I am also still not clear on how their indexing works, but it appears to index through the map functions that are written.. I really need to do something with this data store to understand how it works better).All in all I have used mongo before and love it. I have not used couchdb yet but it could be an alternative if you aren't willing/can't create a java interface for mongodb.Adam Martinek
Posted 3 years ago
Posted 7 months ago
 From the MongoLink Tutorial, applying the format suggested in the section "Modifying Documents" suggests this is the correct syntax: MongoCollectionFind[coll, <|"Age" -> <|"$gt" -> 30|>|>  Answer Posted 7 months ago  This works with mathematica objects (arrays, lists) and with Map for instance but i can't figure out how to use it in cursor objects. If you are doing a MongoDB query, then Mathematica functions (like ToExpression) definitely not supported. The issue is then a MongoDB issue, and there are various options you have (eg see this https://stackoverflow.com/questions/18039358/mongodb-gt-lt-operators-with-prices-stored-as-strings) Answer Posted 3 years ago  Here is a related SE post: mathematica.stackexchange.com/questions/40731 Answer Posted 7 months ago  Thank you all,@Robert Ferguson - You were really quick and concise in you reply The issue is then a MongoDB issue... Indeed but couldn't it be different in future versions of WM? - If the solution to this trivial problem is to convert the source of the data (defeating a little bit the idea of NoSQL way of thinking) or to get hole information to some variable (defeating a bigger bit of the idea of using the cursors) doesn't this situation weakens the all point of using Mongolink on WM?Hope this gets resolved in the version 12.n and in the help menus someone address this issue with examples and some kind of warning, because i'm really enjoying working with WM which was unknown to me just a few weeks ago. Answer Posted 7 months ago  If the solution to this trivial problem is to convert the source of the data (defeating a little bit the idea of NoSQL way of thinking) That is the correct solution if you intend to represent this data in any database and perform database-native queries on the data. NoSQL does not imply 'no types for fields', just no fixed schema. Indeed but couldn't it be different in future versions of WM? How so? How can MongoLink change the behaviour of the MongoDB$gt operator?
 I know MongoLink has to work with the behaviors inherit by MongoDB, but why not create a new function (e.g, named MongoCollectionFindType(coll, {query, typeexpect} )) in which the user say what kind of data it expects to find in the query and if some documents have different types of the expected the function does that conversion in order to make the comparison Because its a deeply non-performant hack to the correct solution: storing the data in the correct format. And if users really insist on storing data with incorrect types, they are free to use MongoCollectionAggregate with the $convert op (https://docs.mongodb.com/manual/reference/operator/aggregation/convert/) if they have MongoDB +4.0 and are willing to pay the appropriate performance penalties. Answer Posted 3 years ago  There is a good presentation with practical code examples about Mathematica with MongoDB here: MongoDB with WL- .NB Presentation Answer Posted 1 year ago  Wolfram Language 11.3 now has support for interfacing with MongoDB: http://reference.wolfram.com/language/MongoLink/guide/MongoLinkOperations.html Answer Posted 1 year ago  That's cool! Thanks for sharing this new feature. Answer Posted 7 months ago  Hello Sebastian,I've put some simple collection with several documents with MongoCollectionInsert. When i try to make a simple query like find the people with 30 years old it works: MongoCollectionFind[coll, <|"Age" -> 30|>] But when i try to know the people with more than 30 years old i cant find the right sintaxe for it.I've tried the mongoDB standard "Age" -> {$gt}->30 sintaxe in all combinations of every other type of brackets and it doesn't work. The examples in the F1 Help only use the equal never the other ones (<,>,<=, >=, !=)
 Well, On youtube i found the answer: MongoCollectionFind[coll, <|"Age" -> |<"$gt"->30>||>] Simple, right? But now i have this problem: - if the "Age" Values are strings (previously "exported" as such by a lazy exporter) the comparison does not work.I've tried using ToExpression applied to the Key "Age" or to Values["Age"] but it doesn't accepted it. This works with mathematica objects (arrays, lists) and with Map for instance but i can't figure out how to use it in cursor objects. Answer Posted 7 months ago  Again, Thank you for your reply. I know MongoLink has to work with the behaviors inherit by MongoDB, but why not create a new function (e.g, named MongoCollectionFindType(coll, {query, typeexpect} )) in which the user say what kind of data it expects to find in the query and if some documents have different types of the expected the function does that conversion in order to make the comparison...and of course this new function does not return a cursor of the documents matching the query, it has to return a list of some sort (maybe not all "columns" but only some chosen by the user), but with the advantage of already having converted to the "typeexpected". Other "problem" is the fact that you, after getting a cursor of the documents return by a MongoFindCollection query, cannot use it as a first argument of a new MongoFindCollection.....refining the query and not handling the data just yet. I'm sorry if i'm not making sense. Answer Posted 7 months ago  My questions were mostly to confirm what i thought....and if there was something basic that i was missing.Again, thank you... specially for having the patience to explain the why's and the if's.In my case, as the model of the data to be exported to MongoDB is also being defined, is important not to miss this issues before the data gets stored and starts to grow. Some of my data comes from machines that "don't care" about this sort of inconsistencies so we have to "pass the comb" before we can work on the data. Answer Posted 7 months ago  is important not to miss this issues before the data gets stored and starts to grow. Use MongoDB schema validation in this case to prevent invalid data from being inserted into the database: https://docs.mongodb.com/manual/core/schema-validation/ Answer Posted 6 months ago  Hi, Using some testing collection i've put MongoDB (using MongoCollectionInsert[]) I've been exploring the MongoLink function MongoCollectionUpdateMany by using the equivalent sintaxe from MongoDB for create/update ($set), for increment ($inc), etc for field and arrays but one problem has come when the field (or sub-fields) are arrays: The MongoCollectionUpdateMany[collection, filter, update] has 3 parameters and the equivalent (i assume) on MongodDB is db.collection.updateMany(filter, update, arraysFilters), although appears to also have 3 parameters in reality it has 4 because you define the coolection "outside" the brackets.....and so it became a problem because that arraysFilters has no place on the MongoLink function. db.students.insert([ { "_id" : 1, "grades" : [ 95, 92, 90 ] }, { "_id" : 2, "grades" : [ 98, 100, 102 ] }, { "_id" : 3, "grades" : [ 95, 110, 100 ] } ]) db.students.updateMany( { grades: {$gte: 100 } }, { $set: { "grades.$[element]" : 100 } }, { arrayFilters: [ { "element": { \$gte: 100 } } ] } ) (because grades is a array its no enought to filter it (first parameter), you have to filter by element of the arrays ans the MongoLink don't have this forth parameter) I may give you some examples but i was hopping that someone has dealt with this issue and give me some help.

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Search for ${{\boldsymbol X}^{0}}$ (Light Boson) Resonance in ${{\boldsymbol e}^{+}}$ ${{\boldsymbol e}^{-}}$ $\rightarrow$ ${{\boldsymbol \gamma}}{{\boldsymbol \gamma}}{{\boldsymbol \gamma}}$ INSPIRE search

The limit is for $\Gamma\mathrm {( {{\mathit X}^{0}} \rightarrow {{\mathit e}^{+}} {{\mathit e}^{-}} )}\cdot{}\Gamma\mathrm {( {{\mathit X}^{0}} \rightarrow {{\mathit \gamma}} {{\mathit \gamma}} {{\mathit \gamma}} )}/\Gamma _{{\mathrm {total}}}$. $\mathit C$ invariance forbids spin-0 ${{\mathit X}^{0}}$ coupling to both ${{\mathit e}^{+}}{{\mathit e}^{-}}$ and ${{\mathit \gamma}}{{\mathit \gamma}}{{\mathit \gamma}}$ .
VALUE ($10^{-3}$ eV) CL% DOCUMENT ID TECN  COMMENT
• • • We do not use the following data for averages, fits, limits, etc. • • •
$<0.2$ 95 1
 1994
CNTR ${\mathit m}_{{{\mathit X}^{0}}}=1.1 - 1.9$ MeV
$<1.0$ 95 2
 1994
CNTR ${\mathit m}_{{{\mathit X}^{0}}}=1.1$ MeV
$<2.5$ 95 2
 1994
CNTR ${\mathit m}_{{{\mathit X}^{0}}}=1.4$ MeV
$<120$ 95 2
 1994
CNTR ${\mathit m}_{{{\mathit X}^{0}}}=1.7$ MeV
$<3.8$ 95 3
 1992
CNTR ${\mathit m}_{{{\mathit X}^{0}}}$= 1.5 MeV
1  VO 1994 looked for ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}{{\mathit \gamma}}$ decaying at rest. The precise limits depend on ${\mathit m}_{{{\mathit X}^{0}}}$. See Fig.$~$2(b) in paper.
2  VO 1994 looked for ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}{{\mathit \gamma}}$ decaying in flight.
3  SKALSEY 1992 also give limits $4.3$ for ${\mathit m}_{{{\mathit X}^{0}}}$ = $1.54$ and $7.5$ for $1.64$ MeV. The spin of ${{\mathit X}^{0}}$ is assumed to be one.
References:
 VO 1994
PR C49 1551 Search for Resonance in Multiphoton Final States from Low Energy ${{\mathit e}^{+}}{{\mathit e}^{-}}$ Scattering
 SKALSEY 1992
PRL 68 456 Search for Three Photon Final State Resonances in Low-Energy ${{\mathit e}^{+}}{{\mathit e}^{-}}$ Scattering


text:
# EPR reasoning with triplet state

jk22
EPRB is a know reasoning about the singlet state.

However if we apply this to the triplet state $$\frac{1}{\sqrt{2}}(\mid+-\rangle+\mid-+\rangle)$$.

we find the following :

suppose we measure spin A with result +, then spin B is - (we see this by the projection on the possible state), however it's a state with total spin 1, since this triplet state is eigenvector of the sum operator (S1+S2)^2.

So we fing that the following is different : measuring spin A and then spin B and making the sum,

or making the measurement of the sum. How does the system know in advance the sum will be taken ?

Gold Member
Though the triplet state gives perfect negative-correlations when you measure in the spin-z (+-) basis, it doesn't do this if you measure in the spin-x or spin-y bases. In those bases, you can get perfect positive correlations.

It's only with the singlet state, that you have perfect anti correlations in all cases.

jk22
Thanks

jk22
Does this mean that, even if the angle between the direction of measurement is 0, the correlation depends on the angle of the measurement direction, and is not involving only the relative angle ?

I remind also : does anyhow the triplet state exist, since those are fermions and that the wf has to be antisymmetric ? Or does the fact that this representation does not include space, that the 2 particle can have same spin because they are located at different places ?

Gold Member
Generally, the correlations depend on both the angle of the measurement direction and the relative angle. There are mixed states whose correlations are maximum in the spin-z basis which exhibit no correlations at all in the spin-x or spin-y bases.

The triplet state does exist for fermions. The quantum state as a whole has to be antisymmetric, which means that if the spinor is symmetric, the wavefunction has to be antisymmetric, so that the total state is antisymmetric (i.e. even function times odd function = odd function).

jk22
Thanx. Coming back to my original post question about difference between EPR and QM.

For the triplet state, if we measure (S1+S2)^2 we obtain S=1 with certainty in quantum mechanics.

Does this mean that we cannot "think" locally by measuring only S1 and deducing S2 ?

Gold Member
Thanx. Coming back to my original post question about difference between EPR and QM.

For the triplet state, if we measure (S1+S2)^2 we obtain S=1 with certainty in quantum mechanics.

Does this mean that we cannot "think" locally by measuring only S1 and deducing S2 ?

If you prepare a pair of particles to be in the triplet $S=1$ eigenstate of the joint measurement operator $(\vec{S}_{1}+\vec{S}_{2})^{2}$, then you already know what you expect to find (in probability) of whatever you measure of this state. That is, once you know the state of a system, you can figure out the probabilities of every conceivable measurement, though what measurement outcome you happen to see is completely random (so far as we have been able to see).

That being said, this state is (quite) entangled, and with the mathematical formalism, you can work out what you would expect to find if you were to also measure $\vec{S}_{2}$ given that you just found out the outcome of measuring $\vec{S}_{1}$. Experimentally, entanglement is something that can only be seen if you look at both particles.

On their own, single particles are just that, single. Without any outside information, there is no way in principle of ever knowing whether the particle you are looking is single or half of an entangled pair. Because of this (side note) there is no way of using entanglement to communicate faster than light.

jk22
Yes we cannot transmit information FTL since we don't know which measurement to apply on the other side.

The question was rather this : If we measure (S1+S2)^2 we get aligned spin, S=1. If we measure S1 we get S2 is in the opposite direction, S=0.

This is a quantum mechanical effect, since the measurement of S1 disturbs the system.

However is it not the case that some delay always happen between the measurement of S1 and S2, I suppose the measurement of (S1+S2)^2 implies a "simultaneous" measurement of S1 and S2, but due to the uncertainty principle (on the source) S1 could happen a bit before S2 (hence giving S=0) ?

Gold Member
If you find the total spin to be 1, and the total z-component of the angular momentum to be zero, you have the entangled triplet state. Measuring in the Z-basis, if we find that the z-comoponents of the spin have opposite signs whenever we look. This does not mean though, that the x and y-components will also have opposite signs whenever we look, in particular, because different components of the angular momentum don't commute with one another.

The issue of what sort of delay there is between measuring one particle, and updating the state of the second particle is still unresolved as it is a point that distinguishes different interpretations of quantum mechanics. Experimentally, the effect seems to be instantaneous, or at least, no one has been able to find an upper speed limit yet (10,000x the speed of light is a lower limit at least)

jk22
I had one more point to ask : is it possible to make a rotation of the triplet space basis to get $$\mid ++\rangle +\mid--\rangle$$ ? But then the ms quantum number won't be defined ?


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# Reducing residual variance in modeling

Modeling is a central part not only of statistical inquiry, but also of everyday human sense-making. We use models as metaphors for the world, in a broader sense. Of course, a model that explains the world better (than some other model) is to be preferred, all other things being equal. In this post, we demonstrate that a more “clever” statistical model reduces the residual variance. It should be noted that this “noise reduction” comes at a cost, however: The model gets more complex; there a more parameters in the model.

First, let’s load some data and usual packages:

library(tidyverse)
library(mosaic)
library(gridExtra)

data(tips, package = "reshape2")

We filter only the extreme groups in order to carve out the effect more lucidly. In addition, let’s compute the residuals and the square residuals:

tips2 <- filter(tips, size %in% c(1, 6)) %>%
mutate(ID = 1:nrow(.),
tip_resid = tip - mean(tip),
group_by(size) %>%
mutate(tip_mean_group = mean(tip))

glimpse(tips2)
## Observations: 8
## Variables: 11
## Groups: size [2]
## $total_bill <dbl> 3.07, 10.07, 7.25, 29.80, 34.30, 27.05, 48.17, 8… ##$ tip             <dbl> 1.00, 1.83, 1.00, 4.20, 6.70, 5.00, 5.00, 1.92
## $sex <fct> Female, Female, Female, Female, Male, Female, Ma… ##$ smoker          <fct> Yes, No, No, No, No, No, No, Yes
## $day <fct> Sat, Thur, Sat, Thur, Thur, Thur, Sun, Fri ##$ time            <fct> Dinner, Lunch, Dinner, Lunch, Lunch, Lunch, Dinn…
## $size <int> 1, 1, 1, 6, 6, 6, 6, 1 ##$ ID              <int> 1, 2, 3, 4, 5, 6, 7, 8
## $tip_resid <dbl> -2.33125, -1.50125, -2.33125, 0.86875, 3.36875, … ##$ tips_resid_quad <dbl> 5.4347266, 2.2537516, 5.4347266, 0.7547266, 11.3…
## \$ tip_mean_group  <dbl> 1.4375, 1.4375, 1.4375, 5.2250, 5.2250, 5.2250, …

tip_mean_group refers to the mean value of the response variable tip for each conditioned group.

Next, we compute a helper data frame that stores the group means only:

tips_sum <- tips2 %>%
group_by(size) %>%
summarise(tip = mean(tip))

Now, let’s plot:

That’s the plot for the simpler model without consideration of subgroups. We model the target variable tip as the grand mean (of the whole data set).

p2 <- ggplot(tips2) +
geom_hline(aes(yintercept = mean(tip))) +
geom_segment(aes(x = ID,
xend = ID,
y = tip,
yend = mean(tip)),
color = "grey80") +
geom_point(aes(x = ID, y = tip)) +
labs(title = "tip ~ 1")
p2

In contrast, consider the residuals when conditioned on the subgroup means:

p1 <- ggplot(tips2) +
geom_hline(data = tips_sum,
aes(yintercept = tip,
color = factor(size))) +
geom_segment(aes(x = ID,
xend = ID,
y = tip,
yend = tip_mean_group),
color = "grey80") +
geom_point(aes(x = ID, y = tip,
color = factor(size))) +
theme(legend.position = "none") +
labs(title = "tip ~ size")
p1

As can be seen, the residuals become smaller in this model: The more complex model, where the target variable is modeled as the subgroup average, explains the data better. This models shows a better fit to the data.

grid.arrange(p2, p1, nrow = 1)


text:
# A water hose flow at 5litre a min,how long will it take to fill a 30litres bucket?.

May 29, 2018

$6$ minutes.

#### Explanation:

• $\text{5 L = 1 minute}$

You are looking for:

• $\text{30 L = ? minute/s}$

In knowing this, you can use ratios to solve.

• $\text{5 L : 1 minute}$
• $\text{30 L : ? minutes}$

Now, multiply the $\text{30 L}$ by $\text{1 minute}$

$\text{30 min" xx "1 L" = "30 min" * "L}$

Now, divide the value by the $\text{5 L}$

("30 min" \ * \ "L") / "5 L" = "6 minutes"

A more simple method to solve this would be to see how many times the number $5$ fits into $30$. So,

$\frac{30}{5} = 6$

Therefore you need $6$ lots of $\text{5 L}$ to fill the bucket. This $6$ 'lots' is actually $6$ 'minutes'.


text:
# 0.1 Table of integrals

 Page 1 / 1

## Basic integrals

1. $\int {u}^{n}\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1$

2. $\int \frac{du}{u}=\text{ln}\phantom{\rule{0.1em}{0ex}}|u|+C$

3. $\int {e}^{u}\phantom{\rule{0.2em}{0ex}}du={e}^{u}+C$

4. $\int {a}^{u}\phantom{\rule{0.2em}{0ex}}du=\frac{{a}^{u}}{\text{ln}\phantom{\rule{0.1em}{0ex}}a}+C$

5. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−cos}\phantom{\rule{0.2em}{0ex}}u+C$

6. $\int \text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sin}\phantom{\rule{0.2em}{0ex}}u+C$

7. $\int {\text{sec}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tan}\phantom{\rule{0.2em}{0ex}}u+C$

8. $\int {\text{csc}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−cot}\phantom{\rule{0.2em}{0ex}}u+C$

9. $\int \text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sec}\phantom{\rule{0.2em}{0ex}}u+C$

10. $\int \text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−csc}\phantom{\rule{0.2em}{0ex}}u+C$

11. $\int \text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u|+C$

12. $\int \text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sin}\phantom{\rule{0.2em}{0ex}}u|+C$

13. $\int \text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u+\text{tan}\phantom{\rule{0.2em}{0ex}}u|+C$

14. $\int \text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{csc}\phantom{\rule{0.2em}{0ex}}u-\text{cot}\phantom{\rule{0.2em}{0ex}}u|+C$

15. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}-{u}^{2}}}={\text{sin}}^{-1}\frac{u}{a}+C$

16. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{a}^{2}+{u}^{2}}=\frac{1}{a}{\text{tan}}^{-1}\frac{u}{a}+C$

17. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}{\text{sec}}^{-1}\frac{u}{a}+C$

## Trigonometric integrals

18. $\int {\text{sin}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}u-\frac{1}{4}\text{sin}\phantom{\rule{0.2em}{0ex}}2u+C$

19. $\int {\text{cos}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}u+\frac{1}{4}\text{sin}\phantom{\rule{0.2em}{0ex}}2u+C$

20. $\int {\text{tan}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tan}\phantom{\rule{0.2em}{0ex}}u-u+C$

21. $\int {\text{cot}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{cot}\phantom{\rule{0.2em}{0ex}}u-u+C$

22. $\int {\text{sin}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{3}\left(2+{\text{sin}}^{2}u\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+C$

23. $\int {\text{cos}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{3}\left(2+{\text{cos}}^{2}u\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+C$

24. $\int {\text{tan}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}{\text{tan}}^{2}u+\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{cos}\phantom{\rule{0.2em}{0ex}}u|+C$

25. $\int {\text{cot}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{2}{\text{cot}}^{2}u-\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sin}\phantom{\rule{0.2em}{0ex}}u|+C$

26. $\int {\text{sec}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}\text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}u+\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u+\text{tan}\phantom{\rule{0.2em}{0ex}}u|+C$

27. $\int {\text{csc}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{2}\text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}u+\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{csc}\phantom{\rule{0.2em}{0ex}}u-\text{cot}\phantom{\rule{0.2em}{0ex}}u|+C$

28. $\int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{n}{\text{sin}}^{n-1}u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+\frac{n-1}{n}\int {\text{sin}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

29. $\int {\text{cos}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n}{\text{cos}}^{n-1}u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+\frac{n-1}{n}\int {\text{cos}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

30. $\int {\text{tan}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n-1}{\text{tan}}^{n-1}u-\int {\text{tan}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

31. $\int {\text{cot}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{-1}{n-1}{\text{cot}}^{n-1}u-\int {\text{cot}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

32. $\int {\text{sec}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n-1}\text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{sec}}^{n-2}u+\frac{n-2}{n-1}\int {\text{sec}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

33. $\int {\text{csc}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{-1}{n-1}\text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{csc}}^{n-2}u+\frac{n-2}{n-1}\int {\text{csc}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

34. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{\text{sin}\left(a-b\right)u}{2\left(a-b\right)}-\frac{\text{sin}\left(a+b\right)u}{2\left(a+b\right)}+C$

35. $\int \text{cos}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{\text{sin}\left(a-b\right)u}{2\left(a-b\right)}+\frac{\text{sin}\left(a+b\right)u}{2\left(a+b\right)}+C$

36. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=-\frac{\text{cos}\left(a-b\right)u}{2\left(a-b\right)}-\frac{\text{cos}\left(a+b\right)u}{2\left(a+b\right)}+C$

37. $\int u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sin}\phantom{\rule{0.2em}{0ex}}u-u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+C$

38. $\int u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{cos}\phantom{\rule{0.2em}{0ex}}u+u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+C$

39. $\int {u}^{n}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}{u}^{n}\text{cos}\phantom{\rule{0.2em}{0ex}}u+n\int {u}^{n-1}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du$

40. $\int {u}^{n}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du={u}^{n}\text{sin}\phantom{\rule{0.2em}{0ex}}u-n\int {u}^{n-1}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du$

41. $\begin{array}{cc}\hfill \int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du& =-\frac{{\text{sin}}^{n-1}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m+1}\phantom{\rule{0.2em}{0ex}}u}{n+m}+\frac{n-1}{n+m}\int {\text{sin}}^{n-2}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m}u\phantom{\rule{0.2em}{0ex}}du\hfill \\ & =\frac{{\text{sin}}^{n+1}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m-1}u}{n+m}+\frac{m-1}{n+m}\int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m-2}u\phantom{\rule{0.2em}{0ex}}du\hfill \end{array}$

## Exponential and logarithmic integrals

42. $\int u{e}^{au}\phantom{\rule{0.2em}{0ex}}du=\frac{1}{{a}^{2}}\left(au-1\right){e}^{au}+C$

43. $\int {u}^{n}{e}^{au}\phantom{\rule{0.2em}{0ex}}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\int {u}^{n-1}{e}^{au}\phantom{\rule{0.2em}{0ex}}du$

44. $\int {e}^{au}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}\left(a\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu-b\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\right)+C$

45. $\int {e}^{au}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}\left(a\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu+b\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\right)+C$

46. $\int \text{ln}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}u-u+C$

47. $\int {u}^{n}\text{ln}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{{\left(n+1\right)}^{2}}\left[\left(n+1\right)\text{ln}\phantom{\rule{0.1em}{0ex}}u-1\right]+C$

48. $\int \frac{1}{u\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}u}\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{ln}\phantom{\rule{0.1em}{0ex}}u|+C$

## Hyperbolic integrals

49. $\int \text{sinh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{cosh}\phantom{\rule{0.2em}{0ex}}u+C$

50. $\int \text{cosh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sinh}\phantom{\rule{0.2em}{0ex}}u+C$

51. $\int \text{tanh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.2em}{0ex}}u+C$

52. $\int \text{coth}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sinh}\phantom{\rule{0.2em}{0ex}}u|+C$

53. $\int \text{sech}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du={\text{tan}}^{-1}|\text{sinh}\phantom{\rule{0.2em}{0ex}}u|+C$

54. $\int \text{csch}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{tanh}\phantom{\rule{0.1em}{0ex}}\frac{1}{2}u|+C$

55. $\int {\text{sech}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tanh}\phantom{\rule{0.2em}{0ex}}u+C$

56. $\int {\text{csch}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{coth}\phantom{\rule{0.2em}{0ex}}u+C$

57. $\int \text{sech}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tanh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{sech}\phantom{\rule{0.2em}{0ex}}u+C$

58. $\int \text{csch}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{coth}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{csch}\phantom{\rule{0.2em}{0ex}}u+C$

## Inverse trigonometric integrals

59. $\int {\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u+\sqrt{1-{u}^{2}}+C$

60. $\int {\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}u-\sqrt{1-{u}^{2}}+C$

61. $\int {\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}u-\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+{u}^{2}\right)+C$

62. $\int u\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-1}{4}{\text{sin}}^{-1}u+\frac{u\sqrt{1-{u}^{2}}}{4}+C$

63. $\int u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-1}{4}{\text{cos}}^{-1}u-\frac{u\sqrt{1-{u}^{2}}}{4}+C$

64. $\int u\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{2}+1}{2}{\text{tan}}^{-1}u-\frac{u}{2}+C$

65. $\int {u}^{n}{\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{sin}}^{-1}u-\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1$

66. $\int {u}^{n}{\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{cos}}^{-1}u+\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1$

67. $\int {u}^{n}{\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{tan}}^{-1}u-\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{1+{u}^{2}}\right],n\ne \text{−}1$

## Integrals involving a2 + u2 , a >0

68. $\int \sqrt{{a}^{2}+{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}+{u}^{2}}+\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

69. $\int {u}^{2}\sqrt{{a}^{2}+{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left({a}^{2}+2{u}^{2}\right)\sqrt{{a}^{2}+{u}^{2}}-\frac{{a}^{4}}{8}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

70. $\int \frac{\sqrt{{a}^{2}+{u}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{a}^{2}+{u}^{2}}-a\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}+{u}^{2}}}{u}|+C$

71. $\int \frac{\sqrt{{a}^{2}+{u}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{u}+\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

72. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}+{u}^{2}}}=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

73. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}+{u}^{2}}}=\frac{u}{2}\left(\sqrt{{a}^{2}+{u}^{2}}\right)-\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

74. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{a}^{2}+{u}^{2}}}=-\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C$

75. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{a}^{2}+{u}^{2}}}=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C$

76. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({a}^{2}+{u}^{2}\right)}^{3\text{/}2}}=\frac{u}{{a}^{2}\sqrt{{a}^{2}+{u}^{2}}}+C$

## Integrals involving u2 − a2 , a >0

77. $\int \sqrt{{u}^{2}-{a}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

78. $\int {u}^{2}\sqrt{{u}^{2}-{a}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left(2{u}^{2}-{a}^{2}\right)\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{4}}{8}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

79. $\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{u}^{2}-{a}^{2}}-a{\text{cos}}^{-1}\frac{a}{|u|}+C$

80. $\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{{u}^{2}-{a}^{2}}}{u}+\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

81. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{u}^{2}-{a}^{2}}}=\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

82. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{u}^{2}-{a}^{2}}}=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}+\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

83. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{u}^{2}-{a}^{2}}}=\frac{\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C$

84. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({u}^{2}-{a}^{2}\right)}^{3\text{/}2}}=\text{−}\frac{u}{{a}^{2}\sqrt{{u}^{2}-{a}^{2}}}+C$

## Integrals involving a2 − u2 , a >0

85. $\int \sqrt{{a}^{2}-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C$

86. $\int {u}^{2}\sqrt{{a}^{2}-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left(2{u}^{2}-{a}^{2}\right)\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{4}}{8}{\text{sin}}^{-1}\frac{u}{a}+C$

87. $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{a}^{2}-{u}^{2}}-a\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C$

88. $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{u}\sqrt{{a}^{2}-{u}^{2}}-{\text{sin}}^{-1}\frac{u}{a}+C$

89. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}-{u}^{2}}}=-\frac{u}{u}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C$

90. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C$

91. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{{a}^{2}u}\sqrt{{a}^{2}-{u}^{2}}+C$

92. $\int {\left({a}^{2}-{u}^{2}\right)}^{3\text{/}2}\phantom{\rule{0.2em}{0ex}}du=-\frac{u}{8}\left(2{u}^{2}-5{a}^{2}\right)\sqrt{{a}^{2}-{u}^{2}}+\frac{3{a}^{4}}{8}{\text{sin}}^{-1}\frac{u}{a}+C$

93. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({a}^{2}-{u}^{2}\right)}^{3\text{/}2}}=-\frac{u}{{a}^{2}\sqrt{{a}^{2}-{u}^{2}}}+C$

## Integrals involving 2 au − u2 , a >0

94. $\int \sqrt{2au-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u-a}{2}\sqrt{2au-{u}^{2}}+\frac{{a}^{2}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

95. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{2au-{u}^{2}}}={\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

96. $\int u\sqrt{2au-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-au-3{a}^{2}}{6}\sqrt{2au-{u}^{2}}+\frac{{a}^{3}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

97. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{2au-{u}^{2}}}=-\frac{\sqrt{2au-{u}^{2}}}{au}+C$

## Integrals involving a + bu , a ≠ 0

98. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{{b}^{2}}\left(a+bu-a\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right)+C$

99. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{2{b}^{3}}\left[{\left(a+bu\right)}^{2}-4a\left(a+bu\right)+2{a}^{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right]+C$

100. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\left(a+bu\right)}=\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{u}{a+bu}|+C$

101. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\left(a+bu\right)}=-\frac{1}{au}+\frac{b}{{a}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+bu}{u}|+C$

102. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{{\left(a+bu\right)}^{2}}=\frac{a}{{b}^{2}\left(a+bu\right)}+\frac{1}{{b}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|+C$

103. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{u\phantom{\rule{0.2em}{0ex}}{\left(a+bu\right)}^{2}}=\frac{1}{a\left(a+bu\right)}-\frac{1}{{a}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+bu}{u}|+C$

104. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{{\left(a+bu\right)}^{2}}=\frac{1}{{b}^{3}}\left(a+bu-\frac{{a}^{2}}{a+bu}-2a\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right)+C$

105. $\int u\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du=\frac{2}{15{b}^{2}}\left(3bu-2a\right){\left(a+bu\right)}^{3\text{/}2}+C$

106. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2}{3{b}^{2}}\left(bu-2a\right)\sqrt{a+bu}+C$

107. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2}{15{b}^{3}}\left(8{a}^{2}+3{b}^{2}{u}^{2}-4abu\right)\sqrt{a+bu}+C$

108. $\begin{array}{ccc}\hfill \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}& =\frac{1}{\sqrt{a}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}|+C,\hfill & \text{if}\phantom{\rule{0.2em}{0ex}}a>0\hfill \\ & =\frac{2}{\sqrt{\text{−}a}}\text{tan}-1\sqrt{\frac{a+bu}{\text{−}a}}+C,\hfill & \text{if}\phantom{\rule{0.2em}{0ex}}a<0\hfill \end{array}$

109. $\int \frac{\sqrt{a+bu}}{u}\phantom{\rule{0.2em}{0ex}}du=2\sqrt{a+bu}+a\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}$

110. $\int \frac{\sqrt{a+bu}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2}\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}$

111. $\int {u}^{n}\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du=\frac{2}{b\left(2n+3\right)}\left[{u}^{n}{\left(a+bu\right)}^{3\text{/}2}-na\int {u}^{n-1}\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du\right]$

112. $\int \frac{{u}^{n}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2{u}^{n}\sqrt{a+bu}}{b\left(2n+1\right)}-\frac{2na}{b\left(2n+1\right)}\int \frac{{u}^{n-1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}$

113. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{n}\sqrt{a+bu}}=-\frac{\sqrt{a+bu}}{a\left(n-1\right){u}^{n-1}}-\frac{b\left(2n-3\right)}{2a\left(n-1\right)}\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{n-1}\sqrt{a+bu}}$

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
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(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
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text:
# Strategies for solving the Fermi-Hubbard model on near-term quantum computers

@article{Cade2019StrategiesFS,
title={Strategies for solving the Fermi-Hubbard model on near-term quantum computers},
author={Chris Cade and Lana Mineh and Ashley Montanaro and Stasja Stanisic},
journal={arXiv: Quantum Physics},
year={2019}
}
• Published 12 December 2019
• Physics
• arXiv: Quantum Physics
The Fermi-Hubbard model is of fundamental importance in condensed-matter physics, yet is extremely challenging to solve numerically. Finding the ground state of the Hubbard model using variational methods has been predicted to be one of the first applications of near-term quantum computers. Here we carry out a detailed analysis and optimisation of the complexity of variational quantum algorithms for finding the ground state of the Hubbard model, including costs associated with mapping to a real…

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We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian,
• Physics
• 2021
Noisy intermediate-scale quantum (NISQ) computers could solve quantum-mechanical simulation problems that are beyond the capabilities of classical computers. However, NISQ devices experience
• Physics
Quantum
• 2021
It suffices to use O gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when $n=\mathcal{O}\left(\eta^2\right)$.
• Physics, Computer Science
Nature Reviews Physics
• 2021
An overview of the field of Variational Quantum Algorithms is presented and strategies to overcome their challenges as well as the exciting prospects for using them as a means to obtain quantum advantage are discussed.

## References

SHOWING 1-10 OF 72 REFERENCES

• Physics
Quantum Science and Technology
• 2019
A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum
• Physics
• 2016
Many phenomena of strongly correlated materials are encapsulated in the Fermi-Hubbard model whose thermodynamical properties can be computed from its grand canonical potential according to standard
This Article outlines the details about the gate sequence, the measurement scheme and the relevant error mitigation techniques for the implementation of the Hubbard VQE on a NISQ platform, and performs resource estimation on Hubbard VZE using silicon spin qubits as an example platform.
• Physics
Quantum Science and Technology
• 2019
Quantum simulations are bound to be one of the main applications of near-term quantum computers. Quantum chemistry and condensed matter physics are expected to benefit from these technological
• Physics
Nature
• 2017
The experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms is demonstrated, determining the ground-state energy for molecules of increasing size, up to BeH2.
• Physics
• 2016
Quantum computers are the ideal platform for quantum simulations. Given enough coherent operations and qubits, such machines can be leveraged to simulate strongly correlated materials, where
• Physics
Nature
• 2017
It is shown that for gate-defined quantum dots this disorder can be suppressed in a controlled manner and a detailed characterization of the collective Coulomb blockade transition is realized, which is the finite-size analogue of the interaction-driven Mott metal-to-insulator transition.
• Physics
• 2015
The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the
• Physics
• 2015
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square


text:
If 25% of the total number of employees working in Delhi are females, how many employees working in Delhi are males

### Question Asked by a Student from EXXAMM.com Team

Q 2265678565.     Study the following pie chart carefully to answer the questions.
If 25% of the total number of employees working in Delhi
are females, how many employees working in Delhi are
males?
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436

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324

C

438

D

398

E

None of these

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Prealgebra (7th Edition)

I=PRT P= 2000 dollars R = 8% T = 5 years Write 8% as a decimal I=2000$\times$ $\times$ 0.08 $\times$ 5 Multiply I= 800 dollars The simple interest is 800 dollars. total amount = principal + interest total amount= 2000+800= 2800 dollars Total amount after 5 years will be 2800 dollars


text:
JDFTx  1.4.2
Customization

In Basic compilation we indicated [options] for the cmake command, but left them blank because the defaults should work fine in most cases. Here, we'll discuss some common optional packages, features and performance tweaks, and the [options] used to activate them.

## Finding libraries / optional libraries

• If you manually installed some of the dependencies, and/or they happen to be installed in non-standard locations, you can tell cmake where they are using [options]. For example, if you want to use a custom-installed the GNU Scientific Library in /home/user/gsl, add -D GSL_PATH=/home/user/gsl to [options]
• You can use Intel's Math Kernel Library (MKL) to provide FFT, BLAS and LAPACK. Add -D EnableMKL=yes to [options], and additionally specify MKL_PATH as indicated above if MKL is in a non-standard location (besides /opt/intel/mkl)
• You can use MKL to provide BLAS and LAPACK, but still use FFTW for Fourier transforms, by adding -D ForceFFTW=yes to [options]. We find this option to often be more reliable than using the MKL FFTs.
• Set -D ThreadedBlas=yes or no to indicate whether the BLAS library is multithreaded or not. When used with MKL, this will automatically select whether JDFTx links to the sequential or threaded layers of MKL.
• LibXC provides additional exchange-correlation functionals. JDFTx can link to LibXC version >= 3; add -D EnableLibXC=yes to options, and if necessary specify LIBXC_PATH

## Optional compilation flags

• Add -D EnableProfiling=yes to [options] to get summaries of run times per function and memory usage by object type at the end of calculations.
• Adding -D LinkTimeOptimization=yes will enable link-time optimizations (-ipo for the Intel compilers and -flto for the GNU compilers). Note that this significantly slows down the final link step of the build process.
• Add -D StaticLinking=yes to compile JDFTx statically. This is necessary on Windows and is turned on automatically there. It could also be useful on other platforms to compile on one machine and execute on another without the compiler and support libraries installed.
• At the default optimization level, the compiled executable is not locked to specific CPU features. You can enable machine specific optimizations (-march=native on gcc, -fast on icc) by adding -D CompileNative=yes to [options]. Note however that this might cause your executable to be usable only on machines with CPUs of the same or newer generation than the machine it was compiled on. Also, this rarely provides any real performance benefits, because most of the JDFTx execution time is in the BLAS and FFT libraries anyway.

## Changing compilers

The cmake commands in Basic compilation use the default compiler (typically g++) and reasonable optimization flags. Using a different compiler require environment variables rather than [options] passed to cmake. For example, you can use the intel compiler using the command (note bash-specific syntax for environment variables):

CC=icc CXX=icpc cmake [options] ../jdftx-VERSION/jdftx

Make sure the environment variables for the intel compiler (path settings etc.) are loaded before issuing that command (see the compiler documentation / install notes). Of course, you would probably include -D EnableMKL=yes [options] to also use Intel MKL.

Similarly, to use the Clang compiler:

CC=clang CXX=clang++ cmake [options] ../jdftx-VERSION/jdftx

## GPU support

For GPU support, install the CUDA SDK (either from the website, or your package manager, if available) and add -D EnableCUDA=yes to [options]. If you get an unsupported compiler error, comment out the GCC version check from \$CUDA_DIR/include/host_config.h.

Also consider compiling with the following optional flags:

• -D PinnedHostMemory=yes: use page-locked memory on the host (CPU) to speed up memory transfers to the GPU. Make sure the usage limits allow sufficient / unlimited page-locked memory (see eg. ulimit man page).
• -D CudaAwareMPI=yes: If your MPI library supports direct transfers from GPU memory, this flag will speed up MPI data transfers between GPUs.
• Prior to GPU runs, consider setting the environment variable JDFTX_MEMPOOL_SIZE to a large fraction of the memory size in MB / GPU. For example, say "export JDFTX_MEMPOOL_SIZE=4096" (i.e 4 GB) for a GPU with 6 GB memory. This makes a single memory allocation at the start of the run, and then manages memory internally, bypassing expensive cudaMalloc / cudaFree calls.

If you want to run on a GPU, it must be a discrete (not on-board) NVIDIA GPU with compute capability >= 1.3, since that is the minimum for double precision. You may also need to specify -D CUDA_ARCH=compute_xy and -D CUDA_CODE=sm_xy to match the compute architecture x.y of the oldest GPU you want to run on. It currently defaults to compute capability 3.5 which is the most prominent among compute-capable GPUs and Tesla coprocessors available today. See https://developer.nvidia.com/cuda-gpus for compute capabilities of various GPUs.

Note that you will get a real speedup only if your device has a higher memory bandwidth than your CPU/motherboard/RAM combination, since plane-wave DFT is often a memory-bound computation. Also keep in mind that you need a lot of memory on the GPU to actually fit systems of reasonable size (you probably need at least 1-2 GB of GRAM to handle moderate-sized systems).

When you compile with GPU support, extra executables jdftx_gpu, phonon_gpu and wannier_gpu will be generated that will run code almost exclusively on the GPUs, in addition to the regular executables that only run on CPUs.


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# A new flag is to be designed with six vertical stripes using some or

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A new flag is to be designed with six vertical stripes using some or  [#permalink]

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Updated on: 16 May 2017, 07:33
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A new flag is to be designed with six vertical stripes using some or all of the colors yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent stripes have the same color is ?

(1) 12x81
(2) 16 x 192
(3) 20x 125
(4) 24x216
(5) None of the above

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If you appreciate the post then please click +1Kudos

Originally posted by Kritesh on 16 May 2017, 07:27.
Last edited by Bunuel on 16 May 2017, 07:33, edited 1 time in total.
Renamed the topic and edited the question.
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Re: A new flag is to be designed with six vertical stripes using some or  [#permalink]

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16 May 2017, 07:58
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Kritesh wrote:
A new flag is to be designed with six vertical stripes using some or all of the colors yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent stripes have the same color is ?

(1) 12x81
(2) 16 x 192
(3) 20x 125
(4) 24x216
(5) None of the above

Hi

There are 6 stripes and 4 colors for them.

For the fist stripe we can use all 4 colors - no constrains.

For the second, however, we can't choose color already chosen for the first one, hence 4-1 = 3. For the third we can't choose color, chosen for the second, but can choose one used for the first - again 3. Same logic for the rest.

Final result: $$4*3*3*3*3*3 = 4*3^5 = 12*81$$

Re: A new flag is to be designed with six vertical stripes using some or &nbs [#permalink] 16 May 2017, 07:58
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# Rydberg constant

In spectroscopy, the Rydberg constant, symbol ${\displaystyle R_{\infty }}$ for heavy atoms or ${\displaystyle R_{\text{H}}}$ for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants via his Bohr model. As of 2018, ${\displaystyle R_{\infty }}$ and electron spin g-factor are the most accurately measured physical constants.[1]

The constant is expressed for either hydrogen as ${\displaystyle R_{\text{H}}}$, or at the limit of infinite nuclear mass as ${\displaystyle R_{\infty }}$. In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from an atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing an atom from its ground state. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen ${\displaystyle R_{\text{H}}}$ and the Rydberg formula.

In atomic physics, Rydberg unit of energy, symbol Ry, corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom in a simplified Bohr model.[citation needed]

## Value

### Rydberg constant

The CODATA value is[2]

${\displaystyle R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}}$ = 10973731.568160(21) m−1,

where

• ${\displaystyle m_{\text{e}}}$ is the rest mass of the electron,
• ${\displaystyle e}$ is the elementary charge,
• ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space,
• ${\displaystyle h}$ is the Planck constant, and
• ${\displaystyle c}$ is the speed of light in vacuum.

The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron:

${\displaystyle R_{\text{H}}=R_{\infty }{\frac {m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}\approx 1.09678\times 10^{7}{\text{ m}}^{-1},}$

where

• ${\displaystyle m_{\text{e}}}$ is the mass of the electron,
• ${\displaystyle m_{\text{p}}}$ is the mass of the nucleus (a proton).

### Rydberg unit of energy

The Rydberg unit of energy is equivalent to joules[3] and electronvolts[4] in the following manner:

${\displaystyle 1\ {\text{Ry}}\equiv hcR_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{2}}}={\frac {e^{2}}{8\pi \varepsilon _{0}a_{0}}}=2.179\;872\;361\;1035(42)\times 10^{-18}\ {\text{J}}\ =13.605\;693\;122\;994(26)\ {\text{eV}}.}$

### Rydberg frequency

${\displaystyle cR_{\infty }=3.289\;841\;960\;2508(64)\times 10^{15}\ {\text{Hz}}.}$[5]

### Rydberg wavelength

${\displaystyle {\frac {1}{R_{\infty }}}=9.112\;670\;505\;824(17)\times 10^{-8}\ {\text{m}}}$.

The angular wavelength is

${\displaystyle {\frac {1}{2\pi R_{\infty }}}=1.450\;326\;555\;7696(28)\times 10^{-8}\ {\text{m}}}$.

## Occurrence in Bohr model

The Bohr model explains the atomic spectrum of hydrogen (see hydrogen spectral series) as well as various other atoms and ions. It is not perfectly accurate, but is a remarkably good approximation in many cases, and historically played an important role in the development of quantum mechanics. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the sun.

In the simplest version of the Bohr model, the mass of the atomic nucleus is considered to be infinite compared to the mass of the electron,[6] so that the center of mass of the system, the barycenter, lies at the center of the nucleus. This infinite mass approximation is what is alluded to with the ${\displaystyle \infty }$ subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see Rydberg formula):

${\displaystyle {\frac {1}{\lambda }}=\mathrm {Ry} \cdot {1 \over hc}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$

where n1 and n2 are any two different positive integers (1, 2, 3, ...), and ${\displaystyle \lambda }$ is the wavelength (in vacuum) of the emitted or absorbed light.

${\displaystyle {\frac {1}{\lambda }}=R_{M}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$

where ${\displaystyle R_{M}={\frac {R_{\infty }}{1+{\frac {m_{\text{e}}}{M}}}},}$ and M is the total mass of the nucleus. This formula comes from substituting the reduced mass of the electron.

## Precision measurement

The Rydberg constant is one of the most precisely determined physical constants, with a relative standard uncertainty of under 2 parts in 1012.[2] This precision constrains the values of the other physical constants that define it.[7]

Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, and other such effects, the Rydberg constant ${\displaystyle R_{\infty }}$ cannot be directly measured at very high accuracy from the atomic transition frequencies of hydrogen alone. Instead, the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium). Detailed theoretical calculations in the framework of quantum electrodynamics are used to account for the effects of finite nuclear mass, fine structure, hyperfine splitting, and so on. Finally, the value of ${\displaystyle R_{\infty }}$ is determined from the best fit of the measurements to the theory.[8]

## Alternative expressions

The Rydberg constant can also be expressed as in the following equations.

${\displaystyle R_{\infty }={\frac {\alpha ^{2}m_{\text{e}}c}{4\pi \hbar }}={\frac {\alpha ^{2}}{2\lambda _{\text{e}}}}={\frac {\alpha }{4\pi a_{0}}}}$

and in energy units

${\displaystyle {\text{Ry}}=hcR_{\infty }={\frac {1}{2}}m_{\text{e}}c^{2}\alpha ^{2}={\frac {1}{2}}{\frac {e^{4}m_{\text{e}}}{(4\pi \varepsilon _{0})^{2}\hbar ^{2}}}={\frac {1}{2}}{\frac {m_{\text{e}}c^{2}r_{\text{e}}}{a_{0}}}={\frac {1}{2}}{\frac {hc\alpha ^{2}}{\lambda _{\text{e}}}}={\frac {1}{2}}hf_{\text{C}}\alpha ^{2}={\frac {1}{2}}\hbar \omega _{\text{C}}\alpha ^{2}={\frac {1}{2m_{\text{e}}}}\left({\dfrac {\hbar }{a_{0}}}\right)^{2}={\frac {1}{2}}{\frac {e^{2}}{(4\pi \varepsilon _{0})a_{0}}}.}$

where

• ${\displaystyle m_{\text{e}}}$ is the electron rest mass,
• ${\displaystyle e}$ is the electric charge of the electron,
• ${\displaystyle h}$ is the Planck constant,
• ${\displaystyle \hbar =h/2\pi }$ is the reduced Planck constant,
• ${\displaystyle c}$ is the speed of light in a vacuum,
• ${\displaystyle \varepsilon _{0}}$ is the electrical field constant (permittivity) of free space,
• ${\displaystyle \alpha ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{\hbar c}}}$ is the fine-structure constant,
• ${\displaystyle \lambda _{\text{e}}=h/m_{\text{e}}c}$ is the Compton wavelength of the electron,
• ${\displaystyle f_{\text{C}}=m_{\text{e}}c^{2}/h}$ is the Compton frequency of the electron,
• ${\displaystyle \omega _{\text{C}}=2\pi f_{\text{C}}}$ is the Compton angular frequency of the electron,
• ${\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{e^{2}m_{\text{e}}}}}$ is the Bohr radius,
• ${\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}}$ is the classical electron radius.

The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4π/α times the Bohr radius of the atom.

The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: ${\displaystyle E_{n}=-hcR_{\infty }/n^{2}}$.

• Rydberg formula, includes a discussion of Rydberg's original discovery
• Related physical constants:

## References

1. ^ Pohl, Randolf; Antognini, Aldo; Nez, François; Amaro, Fernando D.; Biraben, François; Cardoso, João M. R.; Covita, Daniel S.; Dax, Andreas; Dhawan, Satish; Fernandes, Luis M. P.; Giesen, Adolf; Graf, Thomas; Hänsch, Theodor W.; Indelicato, Paul; Julien, Lucile; Kao, Cheng-Yang; Knowles, Paul; Le Bigot, Eric-Olivier; Liu, Yi-Wei; Lopes, José A. M.; Ludhova, Livia; Monteiro, Cristina M. B.; Mulhauser, Françoise; Nebel, Tobias; Rabinowitz, Paul; Dos Santos, Joaquim M. F.; Schaller, Lukas A.; Schuhmann, Karsten; Schwob, Catherine; Taqqu, David (2010). "The size of the proton". Nature. 466 (7303): 213–216. Bibcode:2010Natur.466..213P. doi:10.1038/nature09250. PMID 20613837. S2CID 4424731.
2. ^ a b "2018 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
3. ^ "2018 CODATA Value: Rydberg constant times hc in J". NIST. The NIST Reference on Constants, Units, and Uncertainty. Retrieved 2020-02-06.
4. ^ "2018 CODATA Value: Rydberg constant times hc in eV". NIST. The NIST Reference on Constants, Units, and Uncertainty. Retrieved 2020-02-06.
5. ^ "2018 CODATA Value: Rydberg constant times c in Hz". NIST. The NIST Reference on Constants, Units, and Uncertainty. Retrieved 2020-02-05.
6. ^ Coffman, Moody L. (1965). "Correction to the Rydberg Constant for Finite Nuclear Mass". American Journal of Physics. 33 (10): 820–823. Bibcode:1965AmJPh..33..820C. doi:10.1119/1.1970992.
7. ^ P.J. Mohr, B.N. Taylor, and D.B. Newell (2015), "The 2014 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 7.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants. National Institute of Standards and Technology, Gaithersburg, MD 20899. Link to R, Link to hcR. Published in Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2012). "CODATA recommended values of the fundamental physical constants: 2010". Reviews of Modern Physics. 84 (4): 1527–1605. arXiv:1203.5425. Bibcode:2012RvMP...84.1527M. doi:10.1103/RevModPhys.84.1527. S2CID 103378639""{{cite journal}}: CS1 maint: postscript (link) and Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2012). "CODATA Recommended Values of the Fundamental Physical Constants: 2010". Journal of Physical and Chemical Reference Data. 41 (4): 043109. arXiv:1507.07956. Bibcode:2012JPCRD..41d3109M. doi:10.1063/1.4724320""{{cite journal}}: CS1 maint: postscript (link).
8. ^ Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA recommended values of the fundamental physical constants: 2006". Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.