Editing Clausen function

{{short description|Transcendental single-variable function}}
{{redir-multi|2|Log sine function|Log cosine function|the historically used compound functions|logarithmic sine|and|logarithmic cosine}}
[[File:Mplwp Clausen.svg|thumbnail|Graph of the Clausen function {{math|Cl{{sub|2}}(''θ'')}}]]

In [[mathematics]], the '''Clausen function''', introduced by {{harvs|txt|first=Thomas|last=Clausen|authorlink=Thomas Clausen (mathematician)|year=1832}}, is a [[Transcendental number|transcendental]], special [[Function (mathematics)|function]] of a single variable. It can variously be expressed in the form of a [[definite integral]], a [[trigonometric series]], and various other forms. It is intimately connected with the [[polylogarithm]], [[inverse tangent integral]], [[polygamma function]], [[Riemann zeta function]], [[Dirichlet eta function]], and [[Dirichlet beta function]].

The '''Clausen function of order 2''' – often referred to as ''the'' Clausen function, despite being but one of a class of many – is given by the integral:

:<math>\operatorname{Cl}_2(\varphi)=-\int_0^\varphi \log\left|2\sin\frac{x}{2} \right|\, dx:</math>

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:

:<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>

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 [[logarithm]]ic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of [[hypergeometric series]], summations involving the inverse of the [[central binomial coefficient]], sums of the [[polygamma function]], and [[Dirichlet L-series]].

==Basic properties==

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, then <math>\sin k\pi=0</math>

:<math>\operatorname{Cl}_2(m\pi) =0, \quad m= 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots </math>

It has maxima at <math>\theta = \frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>

:<math>\operatorname{Cl}_2\left(\frac{\pi}{3}+2m\pi \right) =1.01494160 \ldots </math>

and minima at <math>\theta = -\frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>

:<math>\operatorname{Cl}_2\left(-\frac{\pi}{3}+2m\pi \right) =-1.01494160 \ldots </math>

The following properties are immediate consequences of the series definition:

:<math>\operatorname{Cl}_2(\theta+2m\pi) = \operatorname{Cl}_2(\theta) </math>
:<math>\operatorname{Cl}_2(-\theta) = -\operatorname{Cl}_2(\theta) </math>

See {{harvtxt|Lu|Perez|1992}}.

==General definition==
{{multiple image
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 | image1 = Mplwp Standard Clausen.svg
 | alt1 = Standard Clausen functions
 | caption1 = Standard Clausen functions
 | width2 = 220
 | image2 = Mplwp Glaisher-Clausen.svg
 | alt2 = Glaisher-Clausen functions
 | caption2 = Glaisher–Clausen functions
}}

More generally, one defines the two generalized Clausen functions:

:<math>\operatorname{S}_z(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta}{k^z}</math>
:<math>\operatorname{C}_z(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta}{k^z}</math>

which are valid for complex ''z'' with Re ''z'' &gt;1. The definition may be extended to all of the complex plane through [[analytic continuation]].

When ''z'' is replaced with a non-negative integer, the '''standard Clausen functions''' are defined by the following [[Fourier series]]:

:<math>\operatorname{Cl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}</math>
:<math>\operatorname{Cl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}</math>
:<math>\operatorname{Sl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}</math>
:<math>\operatorname{Sl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}</math>

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).

==Relation to the Bernoulli polynomials==

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:

:<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>
:<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>

Setting <math>\, x= \theta/2\pi \, </math> in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:

:<math>\operatorname{Sl}_{2m}(\theta) = \frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!} B_{2m}\left(\frac{\theta}{2\pi}\right),</math>
:<math>\operatorname{Sl}_{2m-1}(\theta) = \frac{(-1)^{m}(2\pi)^{2m-1}}{2(2m-1)!} B_{2m-1}\left(\frac{\theta}{2\pi}\right), </math>

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:

:<math>B_n(x)=\sum_{j=0}^n\binom{n}{j} B_jx^{n-j}.</math>

Explicit evaluations derived from the above include:

:<math> \operatorname{Sl}_1(\theta)= \frac{\pi}{2}-\frac \theta 2, </math>
:<math> \operatorname{Sl}_2(\theta)= \frac{\pi^2}{6}-\frac{\pi\theta} 2 +\frac{\theta^2}{4},  </math>
:<math> \operatorname{Sl}_3(\theta)= \frac{\pi^2\theta}{6} -\frac{\pi\theta^2}{4}+\frac{\theta^3}{12},  </math>
:<math> \operatorname{Sl}_4(\theta)= \frac{\pi^4}{90}-\frac{\pi^2\theta^2}{12}+\frac{\pi\theta^3}{12}-\frac{\theta^4}{48}. </math>

==Duplication formula==

For <math> 0 < \theta < \pi </math>, the duplication formula can be proven directly from the integral definition (see also {{harvtxt|Lu|Perez|1992}} for the result – although no proof is given):

:<math>\operatorname{Cl}_2(2\theta) = 2\operatorname{Cl}_2(\theta) - 2\operatorname{Cl}_2(\pi-\theta) </math>

Denoting [[Catalan's constant]] by <math>K=\operatorname{Cl}_2\left(\frac{\pi}{2}\right)</math>, immediate consequences of the duplication formula include the relations:

:<math>\operatorname{Cl}_2\left(\frac{\pi}{4}\right)- \operatorname{Cl}_2 \left(\frac{3\pi} 4\right)=\frac K 2</math>
:<math>2\operatorname{Cl}_2\left(\frac{\pi}{3}\right)= 3\operatorname{Cl}_2 \left(\frac{2\pi} 3\right)</math>

For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace <math> \, \theta \, </math> with the [[Bound variable|dummy variable]] <math>x</math>, and integrate over the interval <math> \, [0, \theta]. \, </math> Applying the same process repeatedly yields:

:<math>\operatorname{Cl}_3(2\theta) = 4\operatorname{Cl}_3(\theta) + 4\operatorname{Cl}_3(\pi-\theta) </math>
:<math>\operatorname{Cl}_4(2\theta) = 8\operatorname{Cl}_4(\theta) - 8\operatorname{Cl}_4(\pi-\theta) </math>
:<math>\operatorname{Cl}_5(2\theta) = 16\operatorname{Cl}_5(\theta) + 16 \operatorname{Cl}_5(\pi-\theta) </math>
:<math>\operatorname{Cl}_6(2\theta) = 32\operatorname{Cl}_6(\theta) - 32 \operatorname{Cl}_6(\pi-\theta) </math>

And more generally, upon induction on <math>\, m, \; m \ge 1 </math>

:<math>\operatorname{Cl}_{m+1}(2\theta) = 2^m\left[\operatorname{Cl}_{m+1}(\theta) + (-1)^m \operatorname{Cl}_{m+1}(\pi-\theta) \right]</math>

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>

:<math>\operatorname{Cl}_{2m}\left(\frac \pi 2 \right) = 2^{2m-1} \left[\operatorname{Cl}_{2m}\left(\frac{\pi}{4}\right)- \operatorname{Cl}_{2m}\left(\frac{3\pi}{4}\right) \right] = \beta(2m)</math>

Where <math>\, \beta(x) \, </math> is the [[Dirichlet beta function]].

===Proof of the duplication formula===

From the integral definition,

:<math>\operatorname{Cl}_2(2\theta)=-\int_0^{2\theta} \log\left| 2 \sin \frac{x}{2} \right| \,dx</math>

Apply the duplication formula for the [[sine function]], <math>\sin x = 2\sin\frac{x}{2}\cos\frac{x}{2}</math> to obtain

:<math>
\begin{align}
& -\int_0^{2\theta} \log\left| \left(2 \sin \frac{x}{4} \right)\left(2 \cos \frac{x}{4}  \right) \right| \,dx \\
= {} & -\int_0^{2\theta} \log\left| 2 \sin \frac{x}{4} \right| \,dx -\int_0^{2\theta} \log\left| 2 \cos \frac{x}{4} \right| \,dx
\end{align}
</math>

Apply the  substitution <math>x=2y, dx=2\, dy</math> on both integrals:

:<math>
\begin{align}
& -2\int_0^\theta \log\left| 2 \sin \frac{x}{2} \right| \,dx -2\int_0^\theta \log\left| 2 \cos \frac{x}{2} \right| \,dx \\
= {} & 2\, \operatorname{Cl}_2(\theta) -2\int_0^\theta  \log\left| 2 \cos \frac{x}{2} \right| \,dx
\end{align}
</math>

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:

: <math>
\begin{align}
& \cos\left(\frac{\pi-y}{2}\right) = \sin \frac{y}{2} \\
\Longrightarrow \qquad & \operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta) -2\int_0^\theta  \log\left| 2 \cos \frac{x}{2} \right| \,dx \\
= {} & 2\, \operatorname{Cl}_2(\theta) +2\int_{\pi}^{\pi-\theta} \log\left| 2 \sin \frac{y}{2} \right| \,dy \\
= {} & 2\, \operatorname{Cl}_2(\theta) -2\, \operatorname{Cl}_2(\pi-\theta) + 2\, \operatorname{Cl}_2(\pi)
\end{align}
</math>

: <math>\operatorname{Cl}_2(\pi) = 0 \, </math>

Therefore,

: <math>\operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta)-2\, \operatorname{Cl}_2(\pi-\theta)\, . \, \Box </math>

==Derivatives of general-order Clausen functions==

Direct differentiation of the [[Fourier series]] expansions for the Clausen functions give:

:<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>
:<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>
:<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>
:<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>

By appealing to the [[First Fundamental Theorem Of Calculus]], we also have:

:<math>\frac{d}{d\theta}\operatorname{Cl}_2(\theta) = \frac{d}{d\theta} \left[ -\int_0^\theta  \log \left| 2\sin \frac{x}{2}\right| \,dx \, \right] = - \log \left| 2\sin \frac{\theta}{2}\right| = \operatorname{Cl}_1(\theta) </math>

==Relation to the inverse tangent integral==

The [[inverse tangent integral]] is defined on the interval <math>0 < z < 1</math> by

:<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>

It has the following closed form in terms of the Clausen function:

:<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>

===Proof of the inverse tangent integral relation===

From the integral definition of the [[inverse tangent integral]], we have

:<math>\operatorname{Ti}_2(\tan \theta) = \int_0^{\tan \theta}\frac{\tan^{-1}x}{x}\,dx</math>

Performing an integration by parts

:<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>
:<math>\theta \log \tan \theta - \int_0^{\tan \theta}\frac{\log x}{1+x^2}\,dx</math>

Apply the substitution <math>x=\tan y,\, y=\tan^{-1}x,\, dy=\frac{dx}{1+x^2}\,</math> to obtain

:<math>\theta \log \tan \theta - \int_0^\theta \log(\tan y)\,dy</math>

For that last integral, apply the transform :<math>y=x/2,\, dy=dx/2\,</math> to get

:<math>
\begin{align}
& \theta \log \tan \theta - \frac 1 2 \int_0^{2\theta}\log\left(\tan \frac x 2 \right)\,dx \\[6pt]
= {} & \theta \log \tan \theta - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{\sin (x/2) }{\cos (x/2)}\right)\,dx \\[6pt]
= {} & \theta \log \tan \theta - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{2\sin (x/2) }{2\cos (x/2)}\right)\,dx \\[6pt]
= {} & \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 \\[6pt]
= {} & \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.
\end{align}
</math>

Finally, as with the proof of the Duplication formula, the substitution <math>x=(\pi-y)\, </math> reduces that last integral to

:<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>

Thus

:<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>

==Relation to the Barnes' G-function==

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]]:

:<math>\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(1+z)} \right) +2\pi z \log \left( \frac{\pi}{ \sin \pi z } \right) </math>

Or equivalently

:<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 z \log \left( \frac{\pi}{ \sin \pi z } \right) </math>

See {{harvtxt|Adamchik|2003}}.

==Relation to the polylogarithm==

The Clausen functions represent the real and imaginary parts of the polylogarithm, on the [[unit circle]]:

:<math>\operatorname{Cl}_{2m}(\theta) = \Im (\operatorname{Li}_{2m}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 1</math>
:<math>\operatorname{Cl}_{2m+1}(\theta) = \Re (\operatorname{Li}_{2m+1}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 0</math>

This is easily seen by appealing to the series definition of the [[polylogarithm]].

:<math>\operatorname{Li}_n(z)=\sum_{k=1}^\infty \frac{z^k}{k^n} \quad \Longrightarrow  \operatorname{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>

By Euler's theorem,

:<math>e^{i\theta} = \cos \theta +i\sin \theta</math>

and by de Moivre's Theorem ([[De Moivre's formula]])

:<math>(\cos \theta +i\sin \theta)^k= \cos k\theta +i\sin k\theta \quad \Rightarrow \operatorname{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>

Hence

:<math>\operatorname{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}} = \operatorname{Sl}_{2m}(\theta)+i\operatorname{Cl}_{2m}(\theta)</math>
:<math>\operatorname{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}} = \operatorname{Cl}_{2m+1}(\theta)+i\operatorname{Sl}_{2m+1}(\theta)</math>

==Relation to the polygamma function==

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:

:<math>\operatorname{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>

An immediate corollary is this equivalent formula in terms of the Hurwitz zeta function:

:<math>\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\zeta\left(2m,\tfrac{j}{2p}\right)+(-1)^q \zeta\left(2m,\tfrac{j+p}{2p}\right)\right].  </math>

{{Collapse top|title=Proof of the formula}}
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):

:<math>\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{k=1}^\infty \frac{\sin (kq\pi/p)}{k^{2m}} </math>

We split this sum into exactly '''p'''-parts, so that the first series contains all, and only, those terms congruent to <math>\,kp+1,\, </math> the second series contains all terms congruent to <math>\,kp+2,\, </math> etc., up to the final '''p'''-th part, that contain all terms congruent to <math>\,kp+p\, </math>

:<math>
\begin{align}
& \operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right) \\
= {} & \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}} +
\sum_{k=0}^\infty \frac{\sin \left[(kp+3)\frac{q\pi}{p}\right]}{(kp+3)^{2m}} + \cdots \\
& \cdots + \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}} +
\sum_{k=0}^\infty \frac{\sin \left[(kp+p)\frac{q\pi}{p}\right]}{(kp+p)^{2m}}
\end{align}
</math>

We can index these sums to form a double sum:

:<math>
\begin{align}
& \operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^{p} \left\{ \sum_{k=0}^\infty \frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(kp+j)^{2m}} \right\} \\
= {} & \sum_{j=1}^{p} \frac{1}{p^{2m}}\left\{ \sum_{k=0}^\infty \frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(k+(j/p))^{2m}} \right\}
\end{align}
</math>

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:

:<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>
:<math>\sin m\pi \equiv 0, \quad \, \cos m\pi \equiv (-1)^m \quad \Longleftrightarrow m=0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots </math>
:<math>\sin \left[(kp+j)\frac{q\pi}{p}\right]=(-1)^{kq}\sin\frac{qj\pi}{p}</math>

Consequently,

:<math>\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^p \frac{1}{p^{2m}} \sin\left(\frac{qj\pi}{p}\right)\, \left\{ \sum_{k=0}^\infty \frac{(-1)^{kq}}{(k+(j/p))^{2m}} \right\}  </math>

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:

:<math>
\begin{align}
& \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}} \\
= {} & \sum_{k=0}^\infty \frac{1}{(2k+(j/p))^{2m}}+ (-1)^q\, \sum_{k=0}^\infty \frac{1}{(2k+1+(j/p))^{2m}} \\
= {} & \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]
\end{align}
</math>

For <math>\,m \in\mathbb{Z} \ge 1\, </math>, the [[polygamma function]] has the series representation

:<math>\psi_m(z)=(-1)^{m+1}m! \sum_{k=0}^\infty \frac{1}{(k+z)^{m+1}} </math>

So, in terms of the polygamma function, the previous '''inner sum''' becomes:

: <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>

Plugging this back into the '''double sum''' gives the desired result:

:<math>\operatorname{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>

{{Collapse bottom}}

==Relation to the generalized logsine integral==

The '''generalized logsine''' integral is defined by:

:<math>\mathcal{L}s_n^{m}(\theta) = -\int_0^\theta  x^m \log^{n-m-1} \left| 2\sin\frac{x}{2} \right| \, dx</math>

In this generalized notation, the Clausen function can be expressed in the form:

:<math>\operatorname{Cl}_2(\theta) = \mathcal{L}s_2^{0}(\theta) </math>

==Kummer's relation==

[[Ernst Kummer]] and Rogers give the relation

:<math>\operatorname{Li}_2(e^{i \theta}) = \zeta(2) - \theta(2\pi-\theta)/4 + i\operatorname{Cl}_2(\theta)</math>

valid for <math>0\leq \theta \leq 2\pi</math>.

==Relation to the Lobachevsky function==

The '''Lobachevsky function''' &Lambda; or Л is essentially the same function with a change of variable:

:<math>\Lambda(\theta) = - \int_0^\theta \log|2 \sin(t)| \,dt = \operatorname{Cl}_2(2\theta)/2</math>

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>

==Relation to Dirichlet L-functions==
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.

==Series acceleration==
A [[series acceleration]] for the Clausen function is given by

:<math>\frac{\operatorname{Cl}_2(\theta)} \theta = 
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)^{2n}.
</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]] {{harv|Borwein et al.|2000}}.

==Special values==

Recall the [[Barnes G-function]], the [[Catalan constant|Catalan's constant]] ''K'' and the [[Gieseking manifold|Gieseking constant]] ''V''. Some special values include

:<math>\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=K</math>
:<math>\operatorname{Cl}_2\left(\frac{\pi}{3}\right)=V</math>
:<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)=
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>

In general, from the [[Barnes G-function#Reflection formula 1.0|Barnes G-function reflection formula]],

:<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 z\log\left(\frac{\pi}{\sin\pi z}\right) </math>

Equivalently, using Euler's [[reflection formula]] for the gamma function, then,

:<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 z\log\big(\Gamma(z)\Gamma(1 - z)\big) </math>

==Generalized special values==

Some special values for higher order Clausen functions include

:<math>\operatorname{Cl}_{2m}(0)=\operatorname{Cl}_{2m}(\pi) = \operatorname{Cl}_{2m}(2\pi)=0</math>
:<math>\operatorname{Cl}_{2m}\left(\frac{\pi}{2}\right)=\beta(2m)</math>
:<math>\operatorname{Cl}_{2m+1}(0)=\operatorname{Cl}_{2m+1}(2\pi)=\zeta(2m+1)</math>
:<math>\operatorname{Cl}_{2m+1}(\pi)=-\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>\beta(x)</math> is the [[Dirichlet beta function]], <math>\eta(x)</math> is the [[Dirichlet eta function]] (also called the alternating zeta function), and <math>\zeta(x)</math> is the [[Riemann zeta function]].

==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>

Fourier-analytic methods can be used to find the first moments of the square of the function <math>\operatorname{Cl}_2(x)</math> on the interval <math>[0,\pi]</math>:<ref name='M'>{{cite journal | last1 = István | first1 = Mező | title = Log-sine integrals and alternating Euler sums | journal = [[Acta Mathematica Hungarica]] | year = 2020 | issue = 160 | pages = 45–57 | doi=10.1007/s10474-019-00975-w }}</ref>

:<math>\int_0^\pi \operatorname{Cl}_2^2(x)\,dx=\zeta(4),</math>
:<math>\int_0^\pi t\operatorname{Cl}_2^2(x)\,dx=\frac{221}{90720} \pi^{6}-4 \zeta(\overline{5}, 1)-2 \zeta(\overline{4}, 2),</math>
:<math>\int_0^\pi t^2\operatorname{Cl}_2^2(x)\,dx=-\frac{2}{3} \pi\left[12 \zeta(\overline{5}, 1)+6 \zeta(\overline{4}, 2)-\frac{23}{10080} \pi^{6}\right].</math>

Here <math>\zeta</math> denotes the [[multiple zeta function]].

==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>\, K \,</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}\operatorname{Cl}_2(2\theta)-\theta\log 2</math>
:<math>\int_0^\theta \log(\cos x)\,dx=\tfrac{1}{2}\operatorname{Cl}_2(\pi-2\theta)-\theta\log 2</math>
:<math>\int_0^\theta \log(\tan x)\,dx=-\tfrac{1}{2}\operatorname{Cl}_2(2\theta)-\tfrac{1}{2} \operatorname{Cl}_2(\pi-2\theta)</math>
:<math>\int_0^\theta \log(1+\cos x)\,dx=2\operatorname{Cl}_2(\pi-\theta)-\theta\log 2</math>
:<math>\int_0^\theta \log(1-\cos x)\,dx=-2\operatorname{Cl}_2(\theta)-\theta\log 2</math>
:<math>\int_0^\theta \log(1+\sin x)\,dx=2K-2\operatorname{Cl}_2\left(\frac{\pi}{2}+\theta\right) -\theta\log 2</math>
:<math>\int_0^\theta \log(1-\sin x)\,dx=-2K+2\operatorname{Cl}_2\left(\frac{\pi}{2}-\theta\right)-\theta\log 2</math>

==References==
{{Reflist}}

* {{AS ref|27.8|1005}}
*{{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 }}
* {{cite journal | first1=Van E. | last1=Wood | title=Efficient calculation of Clausen's integral
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* [[Leonard Lewin (telecommunications engineer)|Leonard Lewin]], (Ed.). ''Structural Properties of Polylogarithms'' (1991)  American Mathematical Society, Providence, RI. {{ISBN|0-8218-4532-2}}
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* {{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|doi-access=free }}  
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* {{cite arXiv| first1=Viktor. S. | last1=Adamchik | eprint=math/0308086v1 | title=Contributions to the Theory of the Barnes Function |year=2003}}
* {{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
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* {{cite journal|first1=Jonathan M. |last1=Borwein | first2=Armin |last2= Straub | doi=10.1016/j.jat.2013.07.003| journal=J. Approx. Theory|pages=74–88 | volume=193| year=2013|title=Relations for Nielsen Polylogarithms}}
* {{cite arXiv| first1=R. J. | last1=Mathar | eprint=1309.7504 | title=A C99 implementation of the Clausen sums |year=2013| class=math.NA }}

[[Category:Zeta and L-functions]]