Let f*(z)= Sigma(infinity)(j=0) a(j)*z(j) be a convergent series in which {a(j)*}(j=0)(infinity) are known real numbers. In this paper, by referring to Osler's lemma (Osler, 1975), we obtain explicit forms of the two bivariate series Sigma(infinity)(j=0)a(n)* (j+m) r(j) cos(alpha+j)theta and Sigma(infinity)(j=0)a(n)*(j+m) r(j) sin(alpha+j)theta, where r, theta are real variables, alpha is an element of R, n is an element of N and m is an element of{0,1, . . . , n-1}. With some illustrative examples, we also show how to obtain the explicit form of a trigonometric series when f*(z) is explicitly given. Three new integral formulae are derived in this direction. (C) 2016 Elsevier Ltd. All rights reserved.