Theorem 5.4.   and are entire functions, with    and  .  

Proof of Theorem 5.4.

    The ratio test shows that the radius of convergence for both functions is infinity, so they are entire by 4.17, part (i). Part (iii) of that theorem gives  

              

We leave the proof that as an exercise.

(c) 2006 John H. Mathews, Russell W. Howell