The SL-type Clausen function are polynomials in $${\displaystyle \,\theta \,}$$, and are closely related to the Bernoulli polynomials. … See more

For $${\displaystyle 0<\theta <\pi }$$, the duplication formula can be proven directly from the integral definition (see also Lu & Perez (1992) for the result – although no proof is given): See more

Direct differentiation of the Fourier series expansions for the Clausen functions give:
By appealing to the First Fundamental Theorem Of Calculus, … See more

For real $${\displaystyle 0<z<1}$$, the Clausen function of second order can be expressed in terms of the Barnes G-function and … See more

From the integral definition,
$${\displaystyle \operatorname {Cl} _{2}(2\theta )=-\int _{0}^{2\theta }\log \left|2\sin {\frac {x}{2}}\right|\,dx}$$
Apply the duplication … See more

The inverse tangent integral is defined on the interval $${\displaystyle 0<z<1}$$ by
It has the following … See more

From the integral definition of the inverse tangent integral, we have
Performing an integration by parts See more