Understanding the Proof of Theorem 5.1 in a Complex Analysis Paper

[benorin]

The paper I am reading is http://arxiv.org/PS_cache/math/pdf/0506/0506319.pdf ).

I am trying to follow the proof of Theorem 5.1 [a.k.a. eqn. (53)] on pg. 16. Multiplying (22) by (s-1) and differentiating w.r.t. s at s=1 wouldn't be the same as Corrollary 5.2 since s=1, right? So instead use the limit definition, e.g. using

$$\frac{\partial f(1)}{\partial s} = \lim_{s\rightarrow 1} \frac{f(s)-f(1)}{s-1}$$

to compute the derivative, is this the right direction? How does (41) come into play here?

Thanks in advance for getting sucked far enough into my problem to post a reply

-Ben

 

[shmoe]

I haven't read this very thoroughly, but that looks correct on why cor. 5.2 doesn't apply. (22) was only valid when s was not 1.

The limit is the right idea for the derivative. When you multiply by (s-1) in (22), you get an expression for $$f(s)=(s-1)\Phi(1,s,u)$$ that's valid when s is not 1. At s=1, you define f(1) to be 1 as $$\Phi(1,s,u)$$ has a simple pole there with residue 1. (41) is

$$\lim_{s\rightarrow 1}\frac{(s-1)\Phi(1,s,u)-1}{s-1}$$

which is the derivative of this f(s) at s=1.

 

[benorin]

Thank you shmoe! That makes it vividly clear to me.

-Ben