R. B. Paris
We examine the asymptotic expansion of the Touchard polynomials Tn(z) (also known as the exponential polynomials) for large n and complex values of the variable z. In our treatment |z| may be finite or allowed to be large like O(n). We employ the method of steepest descents to a suitable integral representation of Tn(z) and find that the number of saddle points that contribute to the expansion depends on the values of n and z. Numerical results are given to illustrate the accuracy of the various expansions.