Lobachevsky's Function

In mathematics, the Clausen function, introduced by , 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 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:

:$\operatorname_2(\varphi)=-\int_0^\varphi \log\left, 2\sin\frac \\, dx:$

In the range $0 < \varphi < 2\pi\,$ 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:

:$\operatorname_2(\varphi)=\sum_^\infty \frac = \sin\varphi +\frac+\frac+\frac+ \cdots$

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, 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 $\pi, \,$ since if $k\in \mathbb \,$ is an integer, then $\sin k\pi=0$

:$\operatorname_2(m\pi) =0, \quad m= 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots$

It has maxima at \theta = \frac+2m\pi \quad \in\mathbb/math>

:$\operatorname_2\left(\frac+2m\pi \right) =1.01494160 \ldots$

and minima at \theta = -\frac+2m\pi \quad \in\mathbb/math>

:$\operatorname_2\left(-\frac+2m\pi \right) =-1.01494160 \ldots$

The following properties are immediate consequences of the series definition:

:$\operatorname_2(\theta+2m\pi) = \operatorname_2(\theta)$
:$\operatorname_2(-\theta) = -\operatorname_2(\theta)$

See .

General definition

More generally, one defines the two generalized Clausen functions:

:$\operatorname_z(\theta) = \sum_^\infty \frac$
:$\operatorname_z(\theta) = \sum_^\infty \frac$

which are valid for complex ''z'' with Re ''z'' >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:

:$\operatorname_(\theta) = \sum_^\infty \frac$
:$\operatorname_(\theta) = \sum_^\infty \frac$
:$\operatorname_(\theta) = \sum_^\infty \frac$
:$\operatorname_(\theta) = \sum_^\infty \frac$

N.B. The SL-type Clausen functions have the alternative notation $\operatorname_m(\theta)\,$ 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 $\, \theta\,$, and are closely related to the Bernoulli polynomials. This connection is apparent from the Fourier series representations of the Bernoulli polynomials:

:$B_(x)=\frac \, \sum_^\infty \frac.$
:$B_(x)=\frac \, \sum_^\infty \frac.$

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

:$\operatorname_(\theta) = \frac B_\left(\frac\right),$
:$\operatorname_(\theta) = \frac B_\left(\frac\right),$

where the Bernoulli polynomials $\, B_n(x)\,$ are defined in terms of the Bernoulli numbers $\, B_n \equiv B_n(0)\,$ by the relation:

:$B_n(x)=\sum_^n\binom B_jx^.$

Explicit evaluations derived from the above include:

:$\operatorname_1(\theta)= \frac-\frac \theta 2,$
:$\operatorname_2(\theta)= \frac-\frac 2 +\frac,$
:$\operatorname_3(\theta)= \frac -\frac+\frac,$
:$\operatorname_4(\theta)= \frac-\frac+\frac-\frac.$

Duplication formula

For $0 < \theta < \pi$, the duplication formula can be proven directly from the integral definition (see also for the result – although no proof is given):

:$\operatorname_2(2\theta) = 2\operatorname_2(\theta) - 2\operatorname_2(\pi-\theta)$

Denoting Catalan's constant by $K=\operatorname_2\left(\frac\right)$, immediate consequences of the duplication formula include the relations:

:$\operatorname_2\left(\frac\right)- \operatorname_2 \left(\frac 4\right)=\frac K 2$
:$2\operatorname_2\left(\frac\right)= 3\operatorname_2 \left(\frac 3\right)$

For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace $\, \theta \,$ with the dummy variable $x$, and integrate over the interval \, , \theta \, Applying the same process repeatedly yields:

:$\operatorname_3(2\theta) = 4\operatorname_3(\theta) + 4\operatorname_3(\pi-\theta)$
:$\operatorname_4(2\theta) = 8\operatorname_4(\theta) - 8\operatorname_4(\pi-\theta)$
:$\operatorname_5(2\theta) = 16\operatorname_5(\theta) + 16 \operatorname_5(\pi-\theta)$
:$\operatorname_6(2\theta) = 32\operatorname_6(\theta) - 32 \operatorname_6(\pi-\theta)$

And more generally, upon induction on $\, m, \; m \ge 1$

:\operatorname_(2\theta) = 2^m\left operatorname_(\theta) + (-1)^m \operatorname_(\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 $\, m \in \mathbb \ge 1\,$

:\operatorname_\left(\frac \pi 2 \right) = 2^ \left operatorname_\left(\frac\right)- \operatorname_\left(\frac\right) \right= \beta(2m)

Where $\, \beta(x) \,$ is the Dirichlet beta function.

Proof of the duplication formula

From the integral definition,

:$\operatorname_2(2\theta)=-\int_0^ \log\left, 2 \sin \frac \ \,dx$

Apply the duplication formula for the sine function, $\sin x = 2\sin\frac\cos\frac$ to obtain

:$\begin & -\int_0^ \log\left, \left(2 \sin \frac \right)\left(2 \cos \frac \right) \ \,dx \\ = & -\int_0^ \log\left, 2 \sin \frac \ \,dx -\int_0^ \log\left, 2 \cos \frac \ \,dx \end$

Apply the substitution $x=2y, dx=2\, dy$ on both integrals:

:$\begin & -2\int_0^\theta \log\left, 2 \sin \frac \ \,dx -2\int_0^\theta \log\left, 2 \cos \frac \ \,dx \\ = & 2\, \operatorname_2(\theta) -2\int_0^\theta \log\left, 2 \cos \frac \ \,dx \end$

On that last integral, set $y=\pi-x, \, x= \pi-y, \, dx = -dy$, and use the trigonometric identity $\cos(x-y)=\cos x\cos y - \sin x\sin y$ to show that:

: $\begin & \cos\left(\frac\right) = \sin \frac \\ \Longrightarrow \qquad & \operatorname_2(2\theta)=2\, \operatorname_2(\theta) -2\int_0^\theta \log\left, 2 \cos \frac \ \,dx \\ = & 2\, \operatorname_2(\theta) +2\int_^ \log\left, 2 \sin \frac \ \,dy \\ = & 2\, \operatorname_2(\theta) -2\, \operatorname_2(\pi-\theta) + 2\, \operatorname_2(\pi) \end$

: $\operatorname_2(\pi) = 0 \,$

Therefore,

: $\operatorname_2(2\theta)=2\, \operatorname_2(\theta)-2\, \operatorname_2(\pi-\theta)\, . \, \Box$

Derivatives of general-order Clausen functions

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

:$\frac\operatorname_(\theta) = \frac\sum_^\infty \frac=\sum_^\infty \frac=\operatorname_(\theta)$
:$\frac\operatorname_(\theta) = \frac\sum_^\infty \frac=-\sum_^\infty \frac=-\operatorname_(\theta)$
:$\frac\operatorname_(\theta) = \frac\sum_^\infty \frac= -\sum_^\infty \frac=-\operatorname_ (\theta)$
:$\frac\operatorname_(\theta) = \frac\sum_^\infty \frac=\sum_^\infty \frac=\operatorname_ (\theta)$

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

:\frac\operatorname_2(\theta) = \frac \left 2\sin \frac\ \,dx \, \right= - \log \left, 2\sin \frac\ = \operatorname_1(\theta)

Relation to the inverse tangent integral

The inverse tangent integral is defined on the interval $0 < z < 1$ by

:$\operatorname_2(z)=\int_0^z \frac\,dx = \sum_^\infty (-1)^k \frac$

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

:$\operatorname_2(\tan \theta)= \theta\log(\tan \theta) + \frac \operatorname_2(2\theta) +\frac\operatorname_2(\pi-2\theta)$

Proof of the inverse tangent integral relation

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

:$\operatorname_2(\tan \theta) = \int_0^\frac\,dx$

Performing an integration by parts

:$\int_0^ \frac\,dx= \tan^x\log x \, \Bigg, _0^ - \int_0^ \frac\,dx=$
:$\theta \log \tan \theta - \int_0^\frac\,dx$

Apply the substitution $x=\tan y,\, y=\tan^x,\, dy=\frac\,$ to obtain

:$\theta \log \tan \theta - \int_0^\theta \log(\tan y)\,dy$

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

:
\begin
& \theta \log \tan \theta - \frac 1 2 \int_0^\log\left(\tan \frac x 2 \right)\,dx \\ pt= & \theta \log \tan \theta - \frac\int_0^\log\left(\frac\right)\,dx \\ pt= & \theta \log \tan \theta - \frac\int_0^\log\left(\frac\right)\,dx \\ pt= & \theta \log \tan \theta - \frac\int_0^\log\left(2\sin \frac \right)\,dx+ \frac\int_0^\log\left(2\cos \frac\right)\,dx \\ pt= & \theta \log \tan \theta +\frac\operatorname_2(2\theta)+ \frac \int_0^ \log\left(2\cos \frac\right)\,dx.
\end

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

:$\int_0^\log\left(2\cos \frac\right)\,dx= \operatorname_2(\pi-2\theta) - \operatorname_2(\pi) = \operatorname_2(\pi-2\theta)$

Thus

:$\operatorname_2(\tan \theta) = \theta \log \tan \theta +\frac\operatorname_2(2\theta)+ \frac \operatorname_2(\pi-2\theta)\, . \, \Box$

Relation to the Barnes' G-function

For real $0 < z < 1$, the Clausen function of second order can be expressed in terms of the Barnes G-function and (Euler) Gamma function:

:$\operatorname_(2\pi z) = 2\pi \log \left( \frac \right) +2\pi z \log \left( \frac \right)$

Or equivalently

:$\operatorname_(2\pi z) = 2\pi \log \left( \frac \right) -2\pi \log \Gamma(z)+2\pi z \log \left( \frac \right)$

See .

Relation to the polylogarithm

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

:$\operatorname_(\theta) = \Im (\operatorname_(e^)), \quad m\in\mathbb \ge 1$
:$\operatorname_(\theta) = \Re (\operatorname_(e^)), \quad m\in\mathbb \ge 0$

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

:$\operatorname_n(z)=\sum_^\infty \frac \quad \Longrightarrow \operatorname_n\left(e^\right)=\sum_^\infty \frac= \sum_^\infty \frac$

By Euler's theorem,

:$e^ = \cos \theta +i\sin \theta$

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

:$(\cos \theta +i\sin \theta)^k= \cos k\theta +i\sin k\theta \quad \Rightarrow \operatorname_n\left(e^\right)=\sum_^\infty \frac+ i \, \sum_^\infty \frac$

Hence

:$\operatorname_\left(e^\right)=\sum_^\infty \frac+ i \, \sum_^\infty \frac = \operatorname_(\theta)+i\operatorname_(\theta)$
:$\operatorname_\left(e^\right)=\sum_^\infty \frac+ i \, \sum_^\infty \frac = \operatorname_(\theta)+i\operatorname_(\theta)$

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:

:\operatorname_\left( \frac\right)= \frac \, \sum_^ \sin\left(\tfrac\right)\, \left psi_\left(\tfrac\right)+(-1)^q\psi_\left(\tfrac\right)\right

Let $\,p\,$ and $\,q\,$ be positive integers, such that $\,q/p\,$ is a rational number $\,0 < q/p < 1\,$, then, by the series definition for the higher order Clausen function (of even index):

:$\operatorname_\left( \frac\right)= \sum_^\infty \frac$

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

:$\begin & \operatorname_\left( \frac\right) \\ = & \sum_^\infty \frac + \sum_^\infty \frac + \sum_^\infty \frac + \cdots \\ & \cdots + \sum_^\infty \frac + \sum_^\infty \frac + \sum_^\infty \frac \end$

We can index these sums to form a double sum:

:$\begin & \operatorname_\left( \frac\right)= \sum_^ \left\ \\ = & \sum_^ \frac\left\ \end$

Applying the addition formula for the sine function, $\,\sin(x+y)=\sin x\cos y+\cos x\sin y,\,$ the sine term in the numerator becomes:

:\sin \left kp+j)\frac\right\sin\left(kq\pi+\frac\right)=\sin kq\pi \cos \frac+\cos kq\pi \sin\frac
:$\sin m\pi \equiv 0, \quad \, \cos m\pi \equiv (-1)^m \quad \Longleftrightarrow m=0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots$
:\sin \left kp+j)\frac\right(-1)^\sin\frac

Consequently,

:$\operatorname_\left( \frac\right)= \sum_^p \frac \sin\left(\frac\right)\, \left\$

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:

:
\begin
& \sum_^\infty \frac=\sum_^\infty \frac+ \sum_^\infty \frac \\
= & \sum_^\infty \frac+ (-1)^q\, \sum_^\infty \frac \\
= & \frac\left \sum_^\infty \frac+ (-1)^q\, \sum_^\infty \frac \right\end

For $\,m \in\mathbb \ge 1\,$, the polygamma function has the series representation

:$\psi_m(z)=(-1)^m! \sum_^\infty \frac$

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

: \frac \left psi_\left(\tfrac\right)+(-1)^q\psi_ \left(\tfrac\right)\right

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

:\operatorname_\left( \frac\right)= \frac \, \sum_^ \sin\left(\tfrac\right)\, \left psi_\left(\tfrac\right)+(-1)^q\psi_\left(\tfrac\right)\right

Relation to the generalized logsine integral

The generalized logsine integral is defined by:

:$\mathcals_n^(\theta) = -\int_0^\theta x^m \log^ \left, 2\sin\frac \ \, dx$

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

:$\operatorname_2(\theta) = \mathcals_2^(\theta)$

Kummer's relation

Ernst Kummer and Rogers give the relation

:$\operatorname_2(e^) = \zeta(2) - \theta(2\pi-\theta)/4 + i\operatorname_2(\theta)$

valid for $0\leq \theta \leq 2\pi$.

Relation to the Lobachevsky function

The Lobachevsky function Λ or Л is essentially the same function with a change of variable:

:$\Lambda(\theta) = - \int_0^\theta \log, 2 \sin(t), \,dt = \operatorname_2(2\theta)/2$

though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function

:$\int_0^\theta \log, \sec(t), \,dt = \Lambda(\theta+\pi/2)+\theta\log 2.$

Relation to Dirichlet L-functions

For rational values of $\theta/\pi$ (that is, for $\theta/\pi=p/q$ for some integers ''p'' and ''q''), the function $\sin(n\theta)$ can be understood to represent a periodic orbit of an element in the cyclic group, and thus $\operatorname_s(\theta)$ can be expressed as a simple sum involving the Hurwitz zeta function. This allows relations between certain Dirichlet L-functions to be easily computed.

Series acceleration

A series acceleration for the Clausen function is given by

:$\frac \theta = 1-\log, \theta, + \sum_^\infty \frac \left(\frac \theta \right)^$

which holds for $, \theta, <2\pi$. Here, $\zeta(s)$ is the Riemann zeta function. A more rapidly convergent form is given by

:\frac =
3-\log\left rational zeta series