Euler's integrals and multiple sine functions

Division Values of Multiple Sine Functions

We refine a formula on values of multiple sine functions at division points. As applications we prove a formula on a sum of reciprocal trigonometric values, and obtain multiple modularity of a three variable modular function, which concerns a generalization of the Dedekind η function.

The Multiple Gamma-Functions and the Log-Gamma Integrals

In this paper, which is a companion paper to W , starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of log Γ 1± t . This enables us to locate the genesis of two new functions A1/a and C1/a considered by Srivastava and Choi. We consider the closely related function A(a) and the Hurwitz zeta function, which render t...

Approximation of Quasiconvex Functions, and Lower Semicontinuity of Multiple Integrals

We study semicontinuity of multiple integrals ~ f~,u,Du)dx~ where the vector-valued function u is defined for xE ~C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology 1 N of the Sobolev space H 'P(~;~ ). The ...

Uniform convergence of single and double sine series and integrals

Generalized Log Sine Integrals and the Mordell-tornheim Zeta Values

We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor’s multiple sine functions. It is shown that all the integrals are expressed by the Mordell-Tornheim zeta values at positive integers and that the converse is also true. Moreover, we apply the theory of the integral to obtain various new results for the Mordell-Tornheim zeta values.