Thread: Infinite Product

i am having some difficult to expand e^z - 1 in an infinite product. I was using the identity e^z-1 = 2ie^(z/2)sin(z/2i) but i couldn't really figure it out.

also, similar question im trying to prove an identity
1/(e^z-1) = 1/z - 1/2 + 2z ∑(n=1 to infinite) 1/(z^2+4π^2n^2) by taking logarithmic derivatives in e^z - 1

I do not know what to do after... plz help me

Your approach is quite correct and You can obtain the desired result applying the following theorem...

Weierstrass factorization theorem - Wikipedia, the free encyclopedia

Starting from the identity...

(1)

... and remembering that...

(2)

... setting in (2) and inserting it in (1) You obtain...

(3)

Using (3) You should have no difficulties to resolve the second part of the problem...

Kind regards

thank you so much