Hey guys. My textbook has the following exercise:
I've done parts (i) and (ii), but I'm stuck on part (iii). It looks like it should be easy. Usually when a textbook has as the last part of an exercise the phrase "deduce that..." it's something simple. But I just don't get this one.Originally Posted by Priestley, Intro to Complex Analysis 2nd ed
Any help would be much appreciated!
EDIT: thanks for the latex fix! Now if I can only get this exercise...
You have to start from the 'funny identity'...
(1)
... and then consider the 'infinite products'...
(2)
(3)
Now if You compute using (2) and (3) ...
(4)
... and derive it You arrive at the expression of Your textbook. The details are left to You...
Kind regards
Thanks guys, but I think I'm supposed to use at least one of the following two facts in my proof:
(i) is holomorphic in everywhere except for ,
(ii) Each is a simple pole.
I was thinking that maybe I could show is bounded on . Then I could observe that is holomorphic everywhere, and conclude by Liouville that it is a constant function. Finally, I could plug in zero to show that the constant is 1. But unfortunately I don't know how to show that it's bounded (without first showing that anyway).
I appreciate the suggestions though.
EDIT: I asked my prof, and he says I can use the fact that the identity is valid in , and of course from this the proof is obvious.
Thanks again guys!