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. 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. 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 0 < φ < 2 π {displaystyle 0<varphi <2pi ,} 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 π , {displaystyle pi ,,} since if k ∈ Z {displaystyle kin mathbb {Z} ,} is an integer, then sin ⁡ k π = 0 {displaystyle sin kpi =0} It has maxima at θ = π 3 + 2 m π [ m ∈ Z ] {displaystyle heta ={frac {pi }{3}}+2mpi quad } and minima at θ = − π 3 + 2 m π [ m ∈ Z ] {displaystyle heta =-{frac {pi }{3}}+2mpi quad }