A family of multiple integrals analytically solvable (Q1809030)

Starting from the integral definition of the beta function \(\beta(a,b)\), and performing the analytic continuation in \(a,b\) from the region \(\operatorname {Re} a>-1\), \(\operatorname {Re}b<1\), where the ordinary expression for the integral converges, to the rest of the complex plane, for the two variables \(a\), \(b\), and by further extending the method to multiple integrals of this sort, the author is able to device a procedure that allows him to obtain, in quite an ingenious way, a number of new series and integrals which appear in quantum field theory and statistical physics. The method is somehow reminiscent of the zeta function regularization procedure, but the particular way chosen by the author to perform the analytic continuations actually adds new insights to this field.