In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can 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:

where denotes the natural logarithm 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:

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 is an integer, then

The following properties are immediate consequences of the series definition:

See .

General definition

More generally, one defines the two generalized Clausen functions:

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:

N.B. The SL-type Clausen functions 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, and are closely related to the Bernoulli polynomials. This connection is apparent from the Fourier series representations of the Bernoulli polynomials:

in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:

where the Bernoulli polynomials are defined in terms of the Bernoulli numbers by the relation:

Explicit evaluations derived from the above include:

Duplication formula

, the duplication formula can be proven directly from the series or integral definition (see also for the result – although no proof is given):

Denoting Catalan's constant, immediate consequences of the duplication formula include the relations:

with the dummy variable Applying the same process repeatedly yields:

Use of the generalized duplication formula allows for an extension of the result for the Clausen function of order 2, involving Catalan's constant

is the Dirichlet beta function.

Proof of the duplication formula

From the integral definition,

Apply the duplication formula for the sine function to obtain

on both integrals:

to show that:

Therefore,

Derivatives of general-order Clausen functions

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

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

Relation to the inverse tangent integral

The inverse tangent integral by

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

Proof of the inverse tangent integral relation

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

Performing an integration by parts

to obtain

to get

reduces that last integral to

Thus

Relation to the Barnes' G-function

, the Clausen function of second order can be expressed in terms of the Barnes G-function and (Euler) Gamma function:

Or equivalently

See .

Relation to the polylogarithm

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

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

By Euler's theorem,

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

Hence

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:

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

, then, by the series definition for the higher order Clausen function (of even index):

We split this sum into exactly p etc., up to the final p

We can index these sums to form a double sum:

Applying the addition formula for the sine function the sine term in the numerator becomes:

Consequently,

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:

, the polygamma function has the series representation

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

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

Relation to the generalized logsine integral

The generalized logsine integral is defined by:

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

Kummer's relation

Ernst Kummer and Rogers give the relation

.

Relation to the Lobachevsky function

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

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

Relation to Dirichlet L-functions

for some integers p and q can be understood to represent a periodic orbit of an element in the cyclic group 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

is the Riemann zeta function. A more rapidly convergent form is given by

approaches zero rapidly for large values of n. Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series .

Special values

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

In general, from the Barnes G-function reflection formula,

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

Generalized special values

Some special values for higher order Clausen functions include

is the Dirichlet beta function is the Dirichlet eta function is the Riemann zeta function.

Integrals of the direct function

The following integrals are easily proven from the series representations of the Clausen function:

:

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 (Catalan's constant, and the special cases of the zeta function.

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.

References

• Leonard Lewin, (Ed.). Structural Properties of Polylogarithms (1991) American Mathematical Society, Providence, RI.