Clausen function

In mathematics, the Clausen function, introduced by Thomas Clausen (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, Dirichlet eta function, and Dirichlet beta function.

Graph of the Clausen function Cl2(θ)

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:

In the range 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 propertiesEdit

The Clausen function (of order 2) has simple zeros at all (integer) multiples of   since if   is an integer, then  

 

It has maxima at  

 

and minima at  

 

The following properties are immediate consequences of the series definition:

 
 

(Ref: See Lu and Perez, 1992, below for these results, although no proofs are given).

General definitionEdit

Standard Clausen functions
Glaisher–Clausen functions

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 have the alternative notation   and are sometimes referred to as the Glaisher–Clausen functions (after James Whitbread Lee Glaisher, hence the GL-notation).

Relation to the Bernoulli polynomialsEdit

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

 
 

Setting   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 formulaEdit

For  , 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):

 

Immediate consequences of the duplication formula, along with use of the special value  , include the relations:

 
 

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

 
 
 
 

And more generally, upon induction on  

 

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  

 

Where   is the Dirichlet beta function.

Proof of the duplication formulaEdit

From the integral definition,

 

Apply the duplication formula for the sine function,   to obtain

 

Apply the substitution   on both integrals:

 

On that last integral, set  , and use the trigonometric identity   to show that:

 
 

Therefore,

 

Derivatives of general-order Clausen functionsEdit

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 integralEdit

The inverse tangent integral is defined on the interval   by

 

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

 

Proof of the inverse tangent integral relationEdit

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

 

Performing an integration by parts

 
 

Apply the substitution   to obtain

 

For that last integral, apply the transform :  to get

 

Finally, as with the proof of the Duplication formula, the substitution   reduces that last integral to

 

Thus

 

Relation to the Barnes' G-functionEdit

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

 

Or equivalently

 

Ref: See Adamchik, "Contributions to the Theory of the Barnes function", below.

Relation to the polylogarithmEdit

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 functionEdit

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:

 

Let   and   be positive integers, such that   is a rational number  , then, by the series definition for the higher order Clausen function (of even index):

 

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

 

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:

 

For  , 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 integralEdit

The generalized logsine integral is defined by:

 

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

 

Kummer's relationEdit

Ernst Kummer and Rogers give the relation

 

valid for  .

Relation to the Lobachevsky functionEdit

The Lobachevsky function Λ 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-functionsEdit

For rational values of   (that is, for   for some integers p and q), the function   can be understood to represent a periodic orbit of an element in the cyclic group, and thus   can be expressed as a simple sum involving the Hurwitz zeta function.[citation needed] This allows relations between certain Dirichlet L-functions to be easily computed.

Series accelerationEdit

A series acceleration for the Clausen function is given by

 

which holds for  . Here,   is the Riemann zeta function. A more rapidly convergent form is given by

 

Convergence is aided by the fact that   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, et al., 2000, below).

Special valuesEdit

Recall the Barnes G-function and Catalan constant K. 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 valuesEdit

Some special values for higher order Clausen functions include

 
 
 
 
 

where   is Catalan's constant,   is the Dirichlet beta function,   is the Dirichlet eta function (also called the alternating zeta function), and   is the Riemann zeta function.

 

Integrals of the direct functionEdit

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

 
 
 
 

Fourier-analytic methods can be used to find the first moments of the square of the function   on the interval  :[1]

 
 
 

Here   denotes the Multiple zeta function.

Integral evaluations involving the direct functionEdit

A large number of trigonometric and logarithmo-trigonometric integrals can be evaluated in terms of the Clausen function, and various common mathematical constants like   (Catalan's constant),  , and the special cases of the zeta function,   and  .

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.

 
 
 
 
 
 
 

ReferencesEdit

  1. ^ István, Mező (2020). "Log-sine integrals and alternating Euler sums". Acta Mathematica Hungarica (160): 45–57.