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Residue of f(z) involving digamma function

  1. Jan 14, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the residue of:

    $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$

    Where $n$ is every positive integer because those $n$ are the poles of $f(z)$


    2. Relevant equations


    3. The attempt at a solution

    This is a simple pole, however:

    $$\lim_{z \to n} \frac{(z-n)\psi(-z)}{(z+1)(z+2)^3}$$

    But this gives 0, which is incorrect?
     
  2. jcsd
  3. Jan 14, 2015 #2

    RUber

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    Homework Helper

    For n to be a simple pole of ##\psi (-z) ## you will need to be able to express ##\psi (-z) = (z+n)g(z)## where g(z) is analytic at z=-n. Be careful with your signs.
    Using the expansion ##\psi (n)+\gamma= \sum_{k=1}^n-1 \frac 1k ## shows that for every positive integer n there will be a simple pole at z=-n, since the sum will include k=0.
     
  4. Jan 14, 2015 #3
    Yes I see this, but what about higher order poles? consider:

    Suppose we are given:

    Find the residue at z=n of $$(\psi(-z) + \gamma)^2$$

    Where $$\psi(-z)$$ means the digamma function.

    $$(\psi(-z) +\gamma)^2 = \left( 1 + \frac{1}{2} + \frac{1}{z} + \frac{1}{3} + \frac{1}{z-1} + \frac{1}{4} + \frac{1}{z-2} + ... + \frac{1}{n} + \frac{1}{z-n} + ... \right)\left( 1 + \frac{1}{2} + \frac{1}{z} + \frac{1}{3} + \frac{1}{z-1} + \frac{1}{4} + \frac{1}{z-2} + ... + \frac{1}{n} + \frac{1}{z-n} + ... \right) $$

    We see that we have:

    $$\frac{1}{z-n} \cdot \left( 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} + \frac{1}{z-1} + \frac{1}{z-2} + ... + \frac{1}{z-n} \right) $$

    The pure coefficient of $$\frac{1}{z-n}$$ is simply:

    $$ \left( 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} \right) = H_n$$

    Where $$H_n$$ means the Harmonic Number.

    But then we also had the other $$\frac{1}{z-n}$$ in the other parentheses so twice this.

    $$2 \left( 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} \right) = 2H_n$$

    But does finding the coefficient work for higher order residues?

    Thanks!
     
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