Abstract. An efficient method is presented for the computation of Debye functions of integer orders to twenty significant decimal digits.The Debye functions occur in thermodynamic problems, for example, in the context of crystallographic structure or that of radiation. They are sometimes labeled radiation integrals. For a recent application, refer to Howard and Grindlay [1], and for a survey of numerical tables refer to Fletcher et al. [2]. We shall define these functions as:where Dp(x) + Dp(x) = Ç{p + 1), Riemann's zeta function. Thus the Debye functions are essentially incomplete Riemann zeta functions. Recently Howard and Grindlay [1] used these functions for p = 0.5(0.5)2.5 in the solution of a transcendental equation.Y. L. Luke [3] studied the function A(x, m) = (m m\/xm)Dm(x) for x complex and m an integer. He presented approximations based on the Padé approximation for t/(e' -1), and provided numerical examples for x real, 0 ^ x ^ 10, and m = 1(1)4. He further showed that approximations for higher m can be generated by use of simple recurrence formulas. It should also be observed that, using the basic data given by Luke, it is straightforward to derive approximations for values of m other than an integer, say m an odd multiple of j.In this note we suggest an alternative method for computing Dm(x) and Dm(x) for m = 1(1)10 and 0 Sj (Real x)