![Mplwp Clausen](https://upload.wikimedia.org/wikipedia/commons/c/ce/Mplwp_Clausen.svg)
In
mathematics, the Clausen function, introduced by , is a
transcendental, special
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of a single variable. It can variously be expressed in the form of a
definite integral, a
trigonometric series
In mathematics, a trigonometric series is a infinite series of the form
: \frac+\displaystyle\sum_^(A_ \cos + B_ \sin),
an infinite version of a trigonometric polynomial.
It is called the Fourier series of the integrable function f if the term ...
, and various other forms. It is intimately connected with the
polylogarithm,
inverse tangent integral
The inverse tangent integral is a special function, defined by:
:\operatorname_2(x) = \int_0^x \frac \, dt
Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.
Definition
The invers ...
,
polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
,
Riemann zeta function,
Dirichlet eta function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0:
\eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdo ...
, and
Dirichlet beta function
In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of per ...
.
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
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
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
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
, and
Dirichlet L-series
In mathematics, a Dirichlet ''L''-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...
.
Basic properties
The Clausen function (of order 2) has simple zeros at all (integer) multiples of
since if
is an integer, then
:
It has maxima at
_.