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The problem of expressing an elliptic function in terms of infinite sums of trigonometric functions has been treated by Hermite, Briot and Bouquet, A. C. Dixon and others. In the present paper we treat the same problem from the point of view of Cauchy's residue theorem in function theory, which is also Briot and Bouquet's starting point, but we differ from these authors in that the integrand we use leads to an expansion for an elliptic function which is valid in an arbitrarily wide, but finite, strip of the complex plane, and which contains certain classical results as special cases. An interesting feature of our expansion is that it yields quite directly the Fourier series development of the function. Some examples of this property are indicated as an illustration of the applicability of our formula. It should be noted that the integrand used was first given by F. Gomes Teixeira in another connection.