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Asymptotic behavior of a power series near its branch point

  1. May 1, 2013 #1

    Mute

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    I was reading a paper the other day that made the following claim, and provided no reference for the assertion. I would like to find a reference or figure out how to derive the asymptotic behavior myself.

    The situation is as follows:

    Suppose we have a function ##f(z)##, defined as a power series about ##z=0##:

    $$f(z) = \sum_{k=0}^\infty a_k z^k.$$

    We assume ##f(z)## has a radius of convergence ##|z| \leq 1##, and we can take ##f(1) = 1##. ##f(z)## can have a finite number of well-defined derivatives at ##z=1##, but at some order the derivatives do not exist at that point. Hence, ##f(z)## has a branch point at ##z = 1##. It can be analytically continued for ##|z| > 1##, with a branch cut running along the real axis from 1 to ##\infty##. For context, ##f(z)## is a probability-generating function in the paper I was reading, but I don't think that's relevant other than it sets ##f(1) = 1## (which really isn't all that important anyways).

    The terms ##a_k## decay as ##k^{-\gamma}## at large k, for positive ##\gamma##.

    The paper claims that ##f(z)## has an asymptotic series near ##z=1##,

    $$f(z) \simeq 1 - b(1-z)^{\phi(\gamma)},$$
    where ##b## is a positive constant and ##\phi(\gamma)## is an exponent that depends on the exponent of ##a_k \sim k^{-\gamma}##.

    The actual claim in the paper goes the other direction: if a series ##\sum_k a_k z^k## with radius of convergence 1 has asymptotic behavior ##(1-z)^\phi## near ##z = 1##, then ##a_k \sim k^{-\phi - 1}##. In their claim, the exponent of the decay of ##a_k## is the same as the exponent of ##(1-z)^\phi##, but I think they may be considering a special case. As such, I'm interested in deriving the asymptotic behavior from the series.

    Does anyone know any references on how to do this, or which derive the rresult? I know that the series for the Polylogarithm, ##\mbox{Li}_\gamma(z) = \sum_{k=1}^\infty z^k/k^\gamma##, or at least its analytic continuation, has a power series about ##\mu = 0## for ##z = \exp(\mu)##, but I'm wondering if the result claimed by the paper is actually a general result for all series having coefficients that decay as ##k^{-\gamma}##.

    Thanks!
     
    Last edited: May 1, 2013
  2. jcsd
  3. May 2, 2013 #2

    Mute

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    After some hunting through references and citing articles, I found a related paper which cites this paper: http://prola.aps.org/abstract/PR/v83/i3/p678_1 .

    In that paper, the author derives an asymptotic series for the Polylogarithm, but it looks like his method may apply to other series with finite radius of convergence.

    The basic idea is to compute the Mellin transform of ##f(e^{-\alpha})##, with respect to ##\alpha##. Then, analytically continue the Mellin transform to the complex plane. For the polylogarithm, the transform is ##\Gamma(y)\zeta(\gamma + y)##, where ##y## is the Mellin-transform variable and ##\zeta## the Reimann zeta-function. These already have well-known analytic continuations. Then, using the inverse Mellin transform with the residue theorem, one can generate a series for ##f(e^{-\alpha})##, valid for ##|\alpha| < 2\pi##. The pole from the zeta function contributes a term proportional to ##\alpha^{\gamma-1}##. All other terms are integer powers of ##\alpha##. Hence, the dominant term in the expansion as ##\alpha \rightarrow 0## depends on the value of ##\gamma##.

    The original paper I was following, which used this expansion without reference, was not clear about which ranges of ##\gamma## they were using when they quote various expansions of ##\sum_k a_k z^k##, when ##a_k \sim k^{-\gamma}##. Hence my confusion in the previous post.

    Anywho, maybe this will be useful for someone else in the future, so I figured I'd post the overview of the details and the reference.

    However, if anyone knows any good resources for asymptotic expansions near branch points, that would be appreciated!
     
    Last edited: May 2, 2013
  4. May 4, 2013 #3
    You might be interested in "Analytic Combinatorics" by Flajolet and Sedgewick. Chapter 6 is "Singularity Analysis of Generating Functions".

    Professor Sedgewick has just concluded a free online course using the book on Coursera: "Analytic Combinatorics, Part II".

    https://class.coursera.org/introACpartII-001/class/index
     
  5. May 5, 2013 #4

    Mute

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    Thanks for the reference! I'll check it out when I have a chance.
     
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