Complex exponential proof using power series

[randybryan]

I need to prove that e^z1 x e^z2 = e^{(z1 + z2)}

using the power series e^z = (SUM FROM n=0 to infinity) zⁿ/n!

(For some reason the Sigma operator isn't working)

In the proof I have been given, it reads

(SUM from 0 to infinity) z1ⁿ/n! x (SUM from 0 to infinity)z2^m/m!

= (SUM n,m) z1ⁿz2^m/n!m!

and this is the step that I can't follow:

= (SUM from p=0 to infinity) x (SUM from q=0 to p) z1^q z2^(p-q)/q!(p-q)!

It may be easier to copy out onto paper using the sigma symbols rather than the ridiculous brackets, but I can't get the Sigma operator to work (like I say).

I just don't understand where the q and (p - q) have come from and how it can be split into a multiple of a sum from p=0 to infinity and q=0 to p.

If anyone can explain, I would be extremely grateful :)

= (SUM from p=0 to infinity) 1/p! (SUM from q=0 to p) p!/q! (p- q)! x z1^q x z2^(p-q)

= (SUM from p=0 to infinity) 1/p! (z1 + z2)^p

 

[chiro]

Wouldn't it be easier to prove the LHS from the RHS?

If you do the RHS you can use the binomial theorem. Does that explain where you get those terms from?