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Need help following a proof

  1. Jul 12, 2006 #1

    benorin

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    The paper I am reading is Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent (PDF) by Jesus Guillera & Jonathan Sondow (look here for other formats: http://arxiv.org/abs/math.NT/0506319 ).

    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

    [tex]\frac{\partial f(1)}{\partial s} = \lim_{s\rightarrow 1} \frac{f(s)-f(1)}{s-1} [/tex]

    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
     
  2. jcsd
  3. Jul 12, 2006 #2

    shmoe

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    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 [tex]f(s)=(s-1)\Phi(1,s,u)[/tex] that's valid when s is not 1. At s=1, you define f(1) to be 1 as [tex]\Phi(1,s,u)[/tex] has a simple pole there with residue 1. (41) is

    [tex]\lim_{s\rightarrow 1}\frac{(s-1)\Phi(1,s,u)-1}{s-1}[/tex]

    which is the derivative of this f(s) at s=1.
     
  4. Jul 13, 2006 #3

    benorin

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

    -Ben
     
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