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| [[File:Clausen-function.png|thumbnail|Graph of the Clausen function Cl<sub>2</sub>(θ)]]
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| In [[mathematics]], the '''Clausen function''' - introduced by {{harvs|txt|first=Thomas|last=Clausen|authorlink=Thomas Clausen (mathematician)|year=1832}} - is a transcendental, special function of a single variable. It can variously be expressed in the form of a [[definite integral]], a [[trigonometric series]], and various other special functions. It is intimately connected with the [[Polylogarithm]], [[Inverse tangent integral]], [[Polygamma function]], [[Riemann Zeta function]], [[Eta function]], and [[Dirichlet beta function]].
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| The '''Clausen function of order 2''' - often referred to at ''the'' Clausen function, despite being but one of a class of many - is given by the integral:
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| :<math>\operatorname{Cl}_2(\varphi)=-\int_0^{\varphi} \log\Bigg|2\sin\frac{x}{2} \Bigg|\, dx:</math>
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| In the range :<math>0 < \varphi < 2\pi\, </math> the [[Sine function]] inside the [[absolute value]] sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the [[Fourier series]] representation:
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| :<math>\operatorname{Cl}_2(\varphi)=\sum_{k=1}^{\infty}\frac{\sin k\varphi}{k^2} = \sin\varphi +\frac{\sin 2\varphi}{2^2}+\frac{\sin 3\varphi}{3^2}+\frac{\sin 4\varphi}{4^2}+ \, \cdots </math>
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| The Clausen functions - as a class of functions - feature extensively in many areas of modern mathematical research, particularly in relation to the evaluation of many classes of [[logarithmic]] and Polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of [[Hypergeometric series]], Central [[Binomial]] sums, sums of the [[Polygamma function]], and [[Dirichlet L-series]]. | |
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| ==Basic properties==
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| The '''Clausen function''' (of order 2) has simple zeros at all (integer) multiples of :<math>\pi, \,</math> since if :<math>k\in \mathbb{Z} \, </math> is an integer, :<math>\sin k\pi=0</math>
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| :<math>\text{Cl}_2(m\pi) =0, \quad m= 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots </math>
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| It has maxima at :<math>\theta = \frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>
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| :<math>\text{Cl}_2\left(\frac{\pi}{3}+2m\pi \right) =1.01494160 \cdots </math>
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| and minima at :<math>\theta = -\frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>
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| :<math>\text{Cl}_2\left(-\frac{\pi}{3}+2m\pi \right) =-1.01494160 \cdots </math>
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| The following properties are immediate consequences of the series definition:
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| :<math>\text{Cl}_2(\theta+2m\pi) = \text{Cl}_2(\theta) </math>
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| :<math>\text{Cl}_2(-\theta) = -\text{Cl}_2(\theta) </math>
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| ('''Ref''': See Lu and Perez, 1992, below for these results - although no proofs are given).
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| ==General definition==
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| More generally, one defines the two generalized Clausen functions:
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| :<math>\operatorname{S}_z(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta}{k^z}</math>
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| :<math>\operatorname{C}_z(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta}{k^z}</math>
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| which are valid for complex ''z'' with Re ''z'' >1. The definition may be extended to all of the complex plane through [[analytic continuation]].
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| When ''z'' is replaced with a non-negative integer, the '''Standard Clausen Functions''' are defined by the following [[Fourier series]]:
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| :<math>\operatorname{Cl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}</math>
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| :<math>\operatorname{Cl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}</math>
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| :<math>\operatorname{Sl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}</math>
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| :<math>\operatorname{Sl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}</math>
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| N.B. The '''SL-type Clausen functions''' have the alternative notation :<math>\operatorname{Gl}_m(\theta)\, </math> and are sometimes referred to as the '''Glaisher-Clausen functions''' (after [[James Whitbread Lee Glaisher]], hence the GL-notation).
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| ==Relation to the Bernoulli Polynomials==
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| The '''SL-type Clausen function''' are polynomials in <math>\, \theta\, </math>, and are closely related to the [[Bernoulli polynomials]]. This connection is apparent from the [[Fourier series]] representations of the Bernoulli Polynomials:
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| :<math>B_{2n-1}(x)=\frac{2(-1)^n(2n-1)!}{(2\pi)^{2n-1}} \, \sum_{k=1}^{\infty}\frac{\sin 2\pi kx}{k^{2n-1}}</math>
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| :<math>B_{2n}(x)=\frac{2(-1)^{n-1}(2n)!}{(2\pi)^{2n}} \, \sum_{k=1}^{\infty}\frac{\cos 2\pi kx}{k^{2n}}</math>
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| Setting <math>\, x= \theta/2\pi \, </math> in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:
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| :<math>\text{Sl}_{2m}(\theta) = \frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!} B_{2m}\left(\frac{\theta}{2\pi}\right)</math>
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| :<math>\text{Sl}_{2m-1}(\theta) = \frac{(-1)^{m}(2\pi)^{2m-1}}{2(2m-1)!} B_{2m-1}\left(\frac{\theta}{2\pi}\right)</math>
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| Where the [[Bernoulli polynomials]] <math>\, B_n(x)\,</math> are defined in terms of the [[Bernoulli numbers]] <math>\, B_n \equiv B_n(0)\, </math> by the relation:
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| :<math>B_n(x)=\sum_{j=0}^n\binom{n}{j} B_jx^{n-j}</math>
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| Explicit evaluations derived from the above include:
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| :<math> \text{Sl}_1(\theta)= \frac{\pi}{2}-\frac{\theta}{2} </math>
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| :<math> \text{Sl}_2(\theta)= \frac{\pi^2}{6}-\frac{\pi\theta}{2}+\frac{\theta^2}{4} </math>
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| :<math> \text{Sl}_3(\theta)= \frac{\pi^2\theta}{6} -\frac{\pi\theta^2}{4}+\frac{\theta^3}{12} </math>
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| :<math> \text{Sl}_4(\theta)= \frac{\pi^4}{90}-\frac{\pi^2\theta^2}{12}+\frac{\pi\theta^3}{12}-\frac{\theta^4}{48} </math>
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| ==Duplication formula==
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| For :<math> 0 < \theta < \pi </math>, the duplication formula can be proven directly from the Integral definition (see also Lu and Perez, 1992, below for the result - although no proof is given):
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| :<math>\operatorname{Cl}_{2}(2\theta) = 2\operatorname{Cl}_{2}(\theta) - 2\operatorname{Cl}_{2}(\pi-\theta) </math>
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| Immediate consequences of the duplication formula, along with use of the special value :<math>\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=G</math>, include the relations:
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| :<math>\operatorname{Cl}_2\left(\frac{\pi}{4}\right)- \operatorname{Cl}_2\left(\frac{3\pi}{4}\right)=\frac{G}{2}</math>
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| :<math>2\operatorname{Cl}_2\left(\frac{\pi}{3}\right)= 3\operatorname{Cl}_2\left(\frac{2\pi}{3}\right)</math>
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| For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace <math> \, \theta \, </math> with the [[dummy variable]] <math>\, x \, </math>, and integrate over the interval <math> \, [0, \theta]. \, </math> Applying the same process repeatedly yields:
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| :<math>\operatorname{Cl}_{3}(2\theta) = 4\operatorname{Cl}_{3}(\theta) + 4\operatorname{Cl}_{3}(\pi-\theta) </math>
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| :<math>\operatorname{Cl}_{4}(2\theta) = 8\operatorname{Cl}_{4}(\theta) - 8\operatorname{Cl}_{4}(\pi-\theta) </math>
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| :<math>\operatorname{Cl}_{5}(2\theta) = 16\operatorname{Cl}_{5}(\theta) + 16 \operatorname{Cl}_{5}(\pi-\theta) </math>
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| :<math>\operatorname{Cl}_{6}(2\theta) = 32\operatorname{Cl}_{6}(\theta) - 32 \operatorname{Cl}_{6}(\pi-\theta) </math>
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| And more generally, upon induction on <math>\, m, \, \, m \ge 1 </math>
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| :<math>\operatorname{Cl}_{m+1}(2\theta) = 2^m\Bigg[\operatorname{Cl}_{m+1}(\theta) + (-1)^m \operatorname{Cl}_{m+1}(\pi-\theta) \Bigg]</math>
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| Use of the generalized duplication formula allows for an extension of the result for the Clausen function of order 2 - involving [[Catalan's constant]]. For <math>\, m \in \mathbb{Z} \ge 1\, </math>
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| :<math>\text{Cl}_{2m}\left(\frac{\pi}{2}\right) = 2^{2m-1}\left[\text{Cl}_{2m}\left(\frac{\pi}{4}\right)- \text{Cl}_{2m}\left(\frac{3\pi}{4}\right) \right] = \beta(2m)</math>
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| Where <math>\, \beta(x) \, </math> is the [[Dirichlet beta function]].
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| ==Proof of the Duplication formula==
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| From the integral definition,
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| :<math>\operatorname{Cl}_2(2\theta)=-\int_0^{2\theta} \log\Bigg| 2 \sin \frac{x}{2} \Bigg| \,dx</math> | |
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| Apply the duplication formula for the [[Sine function]], :<math>\sin 2x = 2\sin\frac{x}{2}\cos\frac{x}{2}</math> to obtain
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| :<math>-\int_0^{2\theta} \log\Bigg| \left(2 \sin \frac{x}{4} \right)\left(2 \cos \frac{x}{4} \right) \Bigg| \,dx=</math>
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| :<math>-\int_0^{2\theta} \log\Bigg| 2 \sin \frac{x}{4} \Bigg| \,dx -\int_0^{2\theta} \log\Bigg| 2 \cos \frac{x}{4} \Bigg| \,dx=</math>
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| Apply the substitution <math>x=2y, dx=2\, dy</math> on both integrals
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| <math>\Rightarrow</math>
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| :<math>-2\int_0^{\theta} \log\Bigg| 2 \sin \frac{x}{2} \Bigg| \,dx -2\int_0^{\theta} \log\Bigg| 2 \cos \frac{x}{2} \Bigg| \,dx=</math>
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| :<math>2\, \operatorname{Cl}_2(\theta) -2\int_0^{\theta} \log\Bigg| 2 \cos \frac{x}{2} \Bigg| \,dx</math>
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| On that last integral, set <math>y=\pi-x, \, x= \pi-y, \, dx = -dy</math>, and use the trigonometric identity <math>\cos(x-y)=\cos x\cos y - \sin x\sin y</math> to show that:
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| <math>\cos\left(\frac{\pi-y}{2}\right) = \sin \frac{y}{2} \Rightarrow </math>
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| <math>\operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta) -2\int_0^{\theta} \log\Bigg| 2 \cos \frac{x}{2} \Bigg| \,dx=</math>
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| <math>2\, \operatorname{Cl}_2(\theta) +2\int_{\pi}^{\pi-\theta} \log\Bigg| 2 \sin \frac{y}{2} \Bigg| \,dy= </math>
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| <math>\, \operatorname{Cl}_2(\theta) -2\, \operatorname{Cl}_2(\pi-\theta) + 2\, \operatorname{Cl}_2(\pi)</math>
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| <math>\operatorname{Cl}_2(\pi) = 0</math>
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| Therefore
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| <math>\operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta)-2\, \operatorname{Cl}_2(\pi-\theta)\, . \, \Box </math>
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| ==Derivatives of general order Clausen functions==
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| Direct differentiation of the [[Fourier series]] expansions for the Clausen functions give:
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| :<math>\frac{d}{d\theta}\operatorname{Cl}_{2m+2}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}=\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}=\operatorname{Cl}_{2m+1}(\theta)</math>
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| :<math>\frac{d}{d\theta}\operatorname{Cl}_{2m+1}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}=-\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m}}=-\operatorname{Cl}_{2m}(\theta)</math>
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| :<math>\frac{d}{d\theta}\operatorname{Sl}_{2m+2}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}= -\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}=-\operatorname{Sl}_{2m+1} (\theta)</math>
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| :<math>\frac{d}{d\theta}\operatorname{Sl}_{2m+1}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}=\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m}}=\operatorname{Sl}_{2m} (\theta)</math>
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| By appealing to the [[First Fundamental Theorem Of Calculus]], we also have:
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| :<math>\frac{d}{d\theta}\operatorname{Cl}_2(\theta) = \frac{d}{d\theta} \left[ -\int_0^{\theta} \log \Bigg| 2\sin \frac{x}{2}\Bigg| \,dx \, \right] = - \log \Bigg| 2\sin \frac{\theta}{2}\Bigg| = \operatorname{Cl}_1(\theta) </math>
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| ==Relation to the Inverse Tangent Integral==
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| The [[Inverse tangent integral]] is defined on the interval :<math>0 < z < 1</math> by
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| :<math>\operatorname{Ti}_2(z)=\int_0^z \frac{\tan^{-1}x}{x}\,dx = \sum_{k=0}^{\infty}(-1)^k\frac{z^{2k+1}}{(2k+1)^2}</math>
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| It has the following closed form in terms of the Clausen Function:
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| :<math>\operatorname{Ti}_2(\tan \theta)= \theta\log(\tan \theta) + \frac{1}{2}\operatorname{Cl}_2(2\theta) +\frac{1}{2}\operatorname{Cl}_2(\pi-2\theta)</math>
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| ==Proof of the Inverse Tangent Integral relation==
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| From the integral definition of the [[Inverse tangent integral]], we have
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| :<math>\operatorname{Ti}_2(\tan \theta) = \int_0^{\tan \theta}\frac{\tan^{-1}x}{x}\,dx</math>
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| Performing an integration by parts
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| :<math>\int_0^{\tan \theta}\frac{\tan^{-1}x}{x}\,dx= \tan^{-1}x\log x \, \Bigg|_0^{\tan \theta} - \int_0^{\tan \theta}\frac{\log x}{1+x^2}\,dx=</math>
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| :<math>\theta \log{\tan \theta} - \int_0^{\tan \theta}\frac{\log x}{1+x^2}\,dx</math>
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| Apply the substitution :<math>x=\tan y,\, y=\tan^{-1}x,\, dy=\frac{dx}{1+x^2}\,</math> to obtain
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| :<math>\theta \log{\tan \theta} - \int_0^{\theta}\log(\tan y)\,dy</math>
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| For that last integral, apply the transform :<math>y=x/2,\, dy=dx/2\,</math> to get
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| :<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(\tan \frac{x}{2}\right)\,dx=</math>
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| :<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{\sin (x/2) }{\cos (x/2)}\right)\,dx=</math>
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| :<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{2\sin (x/2) }{2\cos (x/2)}\right)\,dx=</math>
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| :<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(2\sin \frac{x}{2}\right)\,dx+ \frac{1}{2}\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx=</math>
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| :<math>\theta \log{\tan \theta} +\frac{1}{2}\operatorname{Cl}_2(2\theta)+ \frac{1}{2}\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx</math>
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| Finally, as with the proof of the Duplication formula, the substitution <math>x=(\pi-y)\, </math> reduces that last integral to
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| :<math>\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx= \operatorname{Cl}_2(\pi-2\theta)- \operatorname{Cl}_2(\pi) = \operatorname{Cl}_2(\pi-2\theta)</math>
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| Thus
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| :<math>\operatorname{Ti}_2(\tan \theta) = \theta \log{\tan \theta} +\frac{1}{2}\operatorname{Cl}_2(2\theta)+ \frac{1}{2} \operatorname{Cl}_2(\pi-2\theta)\, . \, \Box </math>
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| ==Relation to the Barnes' G-function==
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| For real :<math>0 < z < 1</math>, the Clausen function of second order can be expressed in terms of the [[Barnes G-function]] and (Euler) [[Gamma function]]:
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| :<math>\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(1+z)} \right) -2\pi \log \left( \frac{\sin \pi z}{ \pi } \right) </math>
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| Or equivalently
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| :<math>\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(z)} \right) -2\pi \log \Gamma(z)-2\pi \log \left( \frac{\sin \pi z}{ \pi } \right) </math>
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| Ref: See '''Adamchik''', "Contributions to the Theory of the Barnes function", below.
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| ==Relation to the Polylogarithm==
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| The Clausen functions represent the real and imaginary parts of the Polylogarithm, on the [[Unit Circle]]:
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| :<math>\operatorname{Cl}_{2m}(\theta) = \Im (\operatorname{Li}_{2m}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 1</math>
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| :<math>\operatorname{Cl}_{2m+1}(\theta) = \Re (\operatorname{Li}_{2m+1}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 0</math>
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| This is easily seen by appealing to the series definition of the [[Polylogarithm]].
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| :<math>\text{Li}_n(z)=\sum_{k=1}^{\infty}\frac{z^k}{k^n} \quad \Rightarrow \text{Li}_n\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\left(e^{i\theta}\right)^k}{k^n}= \sum_{k=1}^{\infty}\frac{e^{ik\theta}}{k^n}</math>
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| By Euler's Theorem,
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| :<math>e^{i\theta} = \cos \theta +i\sin \theta</math>
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| and by de Moivre's Theorem ([[DeMoivre's Formula]])
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| :<math>(\cos \theta +i\sin \theta)^k= \cos k\theta +i\sin k\theta \quad \Rightarrow \text{Li}_n\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^n}+ i \, \sum_{k=1}^{\infty}\frac{\sin k\theta}{k^n}</math>
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| Hence
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| :<math>\text{Li}_{2m}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{2m}}+ i \, \sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m}} = \text{Sl}_{2m}(\theta)+i\text{Cl}_{2m}(\theta)</math>
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| :<math>\text{Li}_{2m+1}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{2m+1}}+ i \, \sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m+1}} = \text{Cl}_{2m+1}(\theta)+i\text{Sl}_{2m+1}(\theta)</math>
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| ==Relation to the Polygamma function==
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| The Clausen functions are intimately connected to the [[Polygamma function]]. Indeed, it is possible to express Clausen functions as linear combinations of sine functions and Polygamma functions. One such relation is shown here, and proven below:
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| :<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}(2m-1)!} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right] </math>
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| Let <math>\,p\,</math> and <math>\,q\,</math> be positive integers, such that <math>\,q/p\,</math> is a rational number <math>\,0 < q/p < 1\, </math>, then, by the series definition for the higher order Clausen function (of even index):
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| :<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{k=1}^{\infty}\frac{\sin (kq\pi/p)}{k^{2m}} </math>
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| We split this sum into exactly '''p'''-parts, so that the first series contains all, and only, those terms congruous to <math>\,kp+1,\, </math> the second series contains all terms congruous to <math>\,kp+2,\, </math> etc, up to the final '''p'''-th part, that contain all terms congruous to <math>\,kp+p\, </math>
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| :<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)=</math>
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| :<math>\sum_{k=0}^{\infty}\frac{\sin \left[(kp+1)\frac{q\pi}{p}\right]}{(kp+1)^{2m}} + \sum_{k=0}^{\infty}\frac{\sin \left[(kp+2)\frac{q\pi}{p}\right]}{(kp+2)^{2m}} +
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| \sum_{k=0}^{\infty}\frac{\sin \left[(kp+3)\frac{q\pi}{p}\right]}{(kp+3)^{2m}} + \, \cdots \, </math>
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| :<math>+ \sum_{k=0}^{\infty}\frac{\sin \left[(kp+p-2)\frac{q\pi}{p}\right]}{(kp+p-2)^{2m}} + \sum_{k=0}^{\infty}\frac{\sin \left[(kp+p-1)\frac{q\pi}{p}\right]}{(kp+p-1)^{2m}} +
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| \sum_{k=0}^{\infty}\frac{\sin \left[(kp+p)\frac{q\pi}{p}\right]}{(kp+p)^{2m}}</math>
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| We can index these sums to form a double sum:
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| :<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^{p} \Bigg\{ \sum_{k=0}^{\infty}\frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(kp+j)^{2m}} \Bigg\} =</math>
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| :<math>\sum_{j=1}^{p} \frac{1}{p^{2m}}\Bigg\{ \sum_{k=0}^{\infty}\frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(k+(j/p))^{2m}} \Bigg\} </math>
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| Applying the addition formula for the [[Sine function]], <math>\,\sin(x+y)=\sin x\cos y+\cos x\sin y,\, </math> the sine term in the numerator becomes:
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| :<math>\sin \left[(kp+j)\frac{q\pi}{p}\right]=\sin\left(kq\pi+\frac{qj\pi}{p}\right)=\sin kq\pi \cos \frac{qj\pi}{p}+\cos kq\pi \sin\frac{qj\pi}{p}</math>
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| :<math>\sin m\pi \equiv 0, \quad \, \cos m\pi \equiv (-1)^m \quad \Leftrightarrow m=0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots </math>
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| :<math>\sin \left[(kp+j)\frac{q\pi}{p}\right]=(-1)^{kq}\sin\frac{qj\pi}{p}</math>
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| Consequently,
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| :<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^{p} \frac{1}{p^{2m}} \sin\left(\frac{qj\pi}{p}\right)\, \Bigg\{ \sum_{k=0}^{\infty}\frac{(-1)^{kq}}{(k+(j/p))^{2m}} \Bigg\} </math>
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| To convert the '''inner sum''' in the double sum into a non-alternating sum, split in two in parts in exactly the same way as the earlier sum was split into '''p'''-parts:
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| :<math>\sum_{k=0}^{\infty}\frac{(-1)^{kq}}{(k+(j/p))^{2m}}=\sum_{k=0}^{\infty}\frac{(-1)^{(2k)q}}{((2k)+(j/p))^{2m}}+ \sum_{k=0}^{\infty}\frac{(-1)^{(2k+1)q}}{((2k+1)+(j/p))^{2m}}=</math>
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| :<math>\sum_{k=0}^{\infty}\frac{1}{(2k+(j/p))^{2m}}+ (-1)^q\, \sum_{k=0}^{\infty}\frac{1}{(2k+1+(j/p))^{2m}}=</math>
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| | |
| :<math>\frac{1}{2^p}\left[ \sum_{k=0}^{\infty}\frac{1}{(k+(j/2p))^{2m}}+ (-1)^q\, \sum_{k=0}^{\infty}\frac{1}{(k+\left(\frac{j+p}{2p}\right))^{2m}} \right]</math>
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| | |
| For <math>\,m \in\mathbb{Z} \ge 1\, </math>, the [[Polygamma function]] has the series representation
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| :<math>\psi_{m}(z)=(-1)^{m+1}m! \, \sum_{k=0}^{\infty}\frac{1}{(k+z)^{m+1}} </math>
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| | |
| So, in terms of the Polygamma function, the previous '''inner sum''' becomes:
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| :<math>\frac{1}{2^{2m}(2m-1)!} \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right] </math>
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| | |
| Plugging this back into the '''double sum''' gives the desired result:
| |
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| :<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}(2m-1)!} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right] </math>
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| | |
| ==Relation to the Generalized Logsine Integral==
| |
| | |
| The '''Generalized Logsine''' Integral is defined by:
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| :<math>\mathcal{L}s_n^{m}(\theta) = -\int_0^{\theta} x^m \log^{n-m-1} \Bigg| 2\sin\frac{x}{2} \Bigg| \, dx</math>
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| | |
| In this generalized notation, the Clausen function can be expressed in the form:
| |
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| :<math>\text{Cl}_2(\theta) = \mathcal{L}s_2^{0}(\theta) </math>
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| | |
| ==Kummer's relation==
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| | |
| [[Ernst Kummer]] and Rogers give the relation
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| :<math>\operatorname{Li}_2(e^{i \theta}) = \zeta(2) - \theta(2\pi-\theta)/4 + i\operatorname{Cl}_2(\theta)</math>
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| | |
| valid for <math>0\leq \theta \leq 2\pi</math>.
| |
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| ==Relation to the Lobachevsky function==
| |
| | |
| The '''Lobachevsky function''' Λ or Л is essentially the same function with a change of variable:
| |
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| :<math>\Lambda(\theta) = - \int_0^\theta \log|2 \sin(t)| \,dt = \operatorname{Cl}_2(2\theta)/2</math>
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| | |
| though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function
| |
| | |
| :<math>\int_0^\theta \log| \sec(t)| \,dt = \Lambda(\theta+\pi/2)+\theta\log 2.</math>
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| ==Relation to Dirichlet L-functions==
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| For rational values of <math>\theta/\pi</math> (that is, for <math>\theta/\pi=p/q</math> for some integers ''p'' and ''q''), the function <math>\sin(n\theta)</math> can be understood to represent a periodic orbit of an element in the [[cyclic group]], and thus <math>\operatorname{Cl}_s(\theta)</math> can be expressed as a simple sum involving the [[Hurwitz zeta function]].{{citation needed|date=July 2013}} This allows relations between certain [[Dirichlet L-function]]s to be easily computed.
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| | |
| ==Series acceleration==
| |
| A [[series acceleration]] for the Clausen function is given by
| |
| | |
| :<math>\frac{\operatorname{Cl}_2(\theta)}{\theta} =
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| 1-\log|\theta| +
| |
| \sum_{n=1}^\infty \frac{\zeta(2n)}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^{2n}
| |
| </math>
| |
| | |
| which holds for <math>|\theta|<2\pi</math>. Here, <math>\zeta(s)</math> is the [[Riemann zeta function]]. A more rapidly convergent form is given by
| |
| | |
| :<math>\frac{\operatorname{Cl}_2(\theta)}{\theta} =
| |
| 3-\log\left[|\theta| \left(1-\frac{\theta^2}{4\pi^2}\right)\right]
| |
| -\frac{2\pi}{\theta} \log \left( \frac{2\pi+\theta}{2\pi-\theta}\right)
| |
| +\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^n.
| |
| </math>
| |
| | |
| Convergence is aided by the fact that <math>\zeta(n)-1</math> approaches zero rapidly for large values of ''n''. Both forms are obtainable through the types of resummation techniques used to obtain [[rational zeta series]]. (ref. Borwein, ''etal.'' 2000, below).
| |
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| ==Special values==
| |
| | |
| Some special values include
| |
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| :<math>\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=G</math>
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| | |
| :<math>\operatorname{Cl}_2\left(\frac{\pi}{3}\right)=3\pi \log\left(
| |
| \frac{G\left(\frac{2}{3}\right)}{ G\left(\frac{1}{3}\right)} \right)-3\pi \log
| |
| \Gamma\left(\frac{1}{3}\right)+\pi \log \left(\frac{ 2\pi }{\sqrt{3}}\right)</math>
| |
| | |
| :<math>\operatorname{Cl}_2\left(\frac{2\pi}{3}\right)=2\pi \log\left(
| |
| \frac{G\left(\frac{2}{3}\right)}{ G\left(\frac{1}{3}\right)} \right)-2\pi \log
| |
| \Gamma\left(\frac{1}{3}\right)+\frac{2\pi}{3} \log \left(\frac{ 2\pi
| |
| }{\sqrt{3}}\right)</math>
| |
| | |
| :<math>\operatorname{Cl}_2\left(\frac{\pi}{4}\right)=
| |
| 2\pi\log \left( \frac{G\left(\frac{7}{8}\right)}{G\left(\frac{1}{8}\right)} \right) -2\pi
| |
| \log \Gamma\left(\frac{1}{8}\right)+\frac{\pi}{4}\log \left( \frac{2\pi}{\sqrt{2-\sqrt{2}}}
| |
| \right)</math>
| |
| | |
| :<math>\operatorname{Cl}_2\left(\frac{3\pi}{4}\right)=
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| 2\pi\log \left( \frac{G\left(\frac{5}{8}\right)}{G\left(\frac{3}{8}\right)} \right) -2\pi
| |
| \log \Gamma\left(\frac{3}{8}\right)+\frac{3\pi}{4}\log \left( \frac{2\pi}{\sqrt{2+\sqrt{2}}}
| |
| \right)</math>
| |
| | |
| :<math>\operatorname{Cl}_2\left(\frac{\pi}{6}\right)=
| |
| 2\pi\log \left( \frac{G\left(\frac{11}{12}\right)}{G\left(\frac{1}{12}\right)} \right) -2\pi
| |
| \log \Gamma\left(\frac{1}{12}\right)+\frac{\pi}{6}\log \left( \frac{2\pi \sqrt{2}
| |
| }{\sqrt{3}-1} \right)</math>
| |
| | |
| :<math>\operatorname{Cl}_2\left(\frac{5\pi}{6}\right)=
| |
| 2\pi\log \left( \frac{G\left(\frac{7}{12}\right)}{G\left(\frac{5}{12}\right)} \right) -2\pi
| |
| \log \Gamma\left(\frac{5}{12}\right)+\frac{5\pi}{6}\log \left( \frac{2\pi \sqrt{2}
| |
| }{\sqrt{3}+1} \right)</math>
| |
| | |
| ==Generalized special values==
| |
| | |
| Some special values for higher order Clausen functions include
| |
| | |
| :<math>\operatorname{Cl}_{2m}\left(0\right)=\operatorname{Cl}_{2m}\left(\pi\right)=\operatorname{Cl}_{2m}\left(2\pi\right)=0</math>
| |
| | |
| :<math>\operatorname{Cl}_{2m}\left(\frac{\pi}{2}\right)=\beta(2m)</math>
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| | |
| :<math>\operatorname{Cl}_{2m+1}\left(0\right)=\operatorname{Cl}_{2m+1}\left(2\pi\right)=\zeta(2m+1)</math>
| |
| | |
| :<math>\operatorname{Cl}_{2m+1}\left(\pi\right)=-\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^{2m}}\right)\zeta(2m+1)</math>
| |
| | |
| :<math>\operatorname{Cl}_{2m+1}\left(\frac{\pi}{2}\right)=-\frac{1}{2^{2m+1}}\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^{4m+1}}\right)\zeta(2m+1)</math>
| |
| | |
| where :<math>G = \beta(2)</math> is [[Catalan's constant]], :<math>\beta(x)</math> is the [[Dirichlet beta function]], :<math>\eta(x)</math> is the [[Eta function]] (also called the alternating Zeta function), and :<math>\zeta(x)</math> is the [[Riemann Zeta function]].
| |
| | |
| :<math>\beta(x)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^x}</math>
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| | |
| ==Integrals of the direct function==
| |
| | |
| The following integrals are easily proven from the series representations of the Clausen function: | |
| | |
| :<math>\int_0^{\theta} \operatorname{Cl}_{2m}(x)\,dx=\zeta(2m+1)-\operatorname{Cl}_{2m+1}(\theta)</math>
| |
| | |
| :<math>\int_0^{\theta} \operatorname{Cl}_{2m+1}(x)\,dx=\operatorname{Cl}_{2m+2}(\theta)</math>
| |
| | |
| :<math>\int_0^{\theta} \operatorname{Sl}_{2m}(x)\,dx=\operatorname{Sl}_{2m+1}(\theta)</math>
| |
| | |
| :<math>\int_0^{\theta} \operatorname{Sl}_{2m+1}(x)\,dx=\zeta(2m+2)-\operatorname{Cl}_{2m+2}(\theta)</math>
| |
| | |
| ==Integral evaluations involving the direct function==
| |
| | |
| A large number of trigonometric and logarithmo-trigonometric integrals can be evaluated in terms of the Clausen function, and various common mathematical constants like <math>\, G \,</math> ([[Catalan's constant]]), <math>\, \log 2 \,</math>, and the special cases of the [[Zeta function]], <math>\, \zeta(2) \,</math> and <math>\, \zeta(3) \,</math>.
| |
| | |
| The examples listed below follow directly from the integral representation of the Clausen function, and the proofs require little more than basic trigonometry, integration by parts, and occasional term-by-term integration of the [[Fourier series]] definitions of the Clausen functions.
| |
| | |
| :<math>\int_0^{\theta}\log(\sin x)\,dx=-\tfrac{1}{2}\text{Cl}_2(2\theta)-\theta\log 2</math>
| |
| | |
| :<math>\int_0^{\theta}\log(\cos x)\,dx=\tfrac{1}{2}\text{Cl}_2(\pi-2\theta)-\theta\log 2</math>
| |
| | |
| :<math>\int_0^{\theta}\log(\tan x)\,dx=-\tfrac{1}{2}\text{Cl}_2(2\theta)-\tfrac{1}{2}\text{Cl}_2(\pi-2\theta)</math>
| |
| | |
| :<math>\int_0^{\theta}\log(1+\cos x)\,dx=2\text{Cl}_2(\pi-\theta)-\theta\log 2</math>
| |
| | |
| :<math>\int_0^{\theta}\log(1-\cos x)\,dx=-2\text{Cl}_2(\theta)-\theta\log 2</math>
| |
| | |
| :<math>\int_0^{\theta}\log(1+\sin x)\,dx=2G-2\text{Cl}_2\left(\frac{\pi}{2}+\theta\right)-\theta\log 2</math>
| |
| | |
| :<math>\int_0^{\theta}\log(1-\sin x)\,dx=-2G+2\text{Cl}_2\left(\frac{\pi}{2}-\theta\right)-\theta\log 2</math>
| |
| | |
| ==References==
| |
| | |
| * {{AS ref|27.8|1005}}
| |
| * {{cite arXiv| first1=Viktor. S. | last1=Adamchik | eprint=math/0308086v1 | title=Contributions to the Theory of the Barnes Function}}
| |
| *{{Cite journal | last1=Clausen | first1=Thomas | title=Über die Function sin φ + (1/2<sup>2</sup>) sin 2φ + (1/3<sup>2</sup>) sin 3φ + etc. | url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0008 | year=1832 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=8 | pages=298–300 | ref=harv }}
| |
| * {{cite journal | first1=Van E. | last1=Wood | title=Efficient calculation of Clausen's integral
| |
| |journal=Math. Comp. | year=1968 | volume=22 | number=104 | pages=883–884 | mr=0239733
| |
| |doi = 10.1090/S0025-5718-1968-0239733-9}}
| |
| * Leonard Lewin, (Ed.). ''Structural Properties of Polylogarithms'' (1991) American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
| |
| * {{cite journal| first1=Kurt Siegfried | last1=Kölbig | title=Chebyshev coefficients for the Clausen function Cl<sub>2</sub>(x)
| |
| |journal=J. Comput. Appl. Math. |year=1995 |volume=64 | number=3 |pages=295–297
| |
| |mr=1365432 |doi=10.1016/0377-0427(95)00150-6}}
| |
| * {{cite web|first1=Jonathan M. |last1=Borwein | first2=Armin |last2= Straub | url=http://www.thecarma.net/jon/nielsenrelations.pdf | title=Relations for Nielsen Polylogarithms}}
| |
| * {{cite journal|first1=Jonathan M. |last1=Borwein | first2=David M. |last2= Bradley |first3=Richard E. |last3=Crandall
| |
| |title=Computational Strategies for the Riemann Zeta Function
| |
| |journal=J. Comp. App. Math.
| |
| |year=2000
| |
| |volume=121 | mr=1780051
| |
| |pages=247–296|ref=harv
| |
| |url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf}}
| |
| * {{cite journal|first1=Mikahil Yu. | last1=Kalmykov | first2=A. | last2=Sheplyakov
| |
| |title=LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine integral
| |
| |journal=Comput. Phys. Comm. |year=2005 | volume=172 | pages=45–59
| |
| |doi=10.1016/j.cpc.2005.04.013 }} {{arxiv| archive=hep-ph | id=0411100}}
| |
| * {{cite arXiv| first1=R. J. | last1=Mathar | eprint=1309.7504 | title=A C99 implementation of the Clausen sums}}
| |
| * {{cite web| first1=Hung Jung | last1=Lu | first2=Christopher A. | last2=Perez |url=http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-5809.pdf | title=Massless one-loop scalar three-point integral and associated Clausen, Glaisher, and L-functions | year=1992}}
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| [[Category:Zeta and L-functions]]
| |