On a functional equation for the exponential function of a complex variable

The following result is well known in the theory of analytic functions; see [1]. Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equation where z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.