Clausen function
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.
The Clausen function (of order 2) has simple zeros at all (integer) multiples of
The following properties are immediate consequences of the series definition: See Lu & Perez (1992).
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.
and are sometimes referred to as the Glaisher–Clausen functions (after James Whitbread Lee Glaisher, hence the GL-notation).
This connection is apparent from the Fourier series representations of the Bernoulli polynomials: Setting
by the relation: Explicit evaluations derived from the above include: For
, the duplication formula can be proven directly from the integral definition (see also Lu & Perez (1992) for the result – although no proof is given): Denoting Catalan's constant by
, immediate consequences of the duplication formula include the relations: For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace
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.
From the integral definition, Apply the duplication formula for the sine function,
to show that: Therefore, Direct differentiation of the Fourier series expansions for the Clausen functions give: By appealing to the First Fundamental Theorem Of Calculus, we also have: The inverse tangent integral is defined on the interval
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 The Clausen functions are intimately connected to the polygamma function.
One such relation is shown here, and proven below: An immediate corollary is this equivalent formula in terms of the Hurwitz zeta function: Let
, 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
etc., up to the final p-th part, that contain all terms congruent to
, 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: The generalized logsine integral is defined by: In this generalized notation, the Clausen function can be expressed in the form: Ernst Kummer and Rogers give the relation valid for
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.
approaches zero rapidly for large values of n. Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series (Borwein et al. 2000).
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, Some special values for higher order Clausen functions include where
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
A large number of trigonometric and logarithmo-trigonometric integrals can be evaluated in terms of the Clausen function, and various common mathematical constants like
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.
