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in a theoretical physics book, the author makes the following claim:

$$\frac{1}{e^z + 1} = \frac{1}{2} + \sum_{n=-\infty}^\infty \frac{1}{(2n+1) i\pi - z}$$ and justifies this as

These series can be derived from a theorem which states that any meromorphic function may be expanded as a summation over its poles and residues at those poles

What's the name of that theorem? It's not really a Laurent series, since the Laurent series is for an expansion around one particular point only. I can see that the poles occur whenever $z = (2n+1)i\pi$ for $n \in \mathbb{N}$, but then where does that constant $1/2$ come from?

EDIT: Well, it appears that the general claim isn't valid, so now I'd be interested in a justification for the expansion in my particular example...

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3 Answers 3

up vote 3 down vote accepted

Mittag-Leffler's theorem guarantees the existence of a meromorphic function $g(z)$ whose poles and principal parts are given by any values specified. Then, if $f(z)$ is a meromorphic function, then $f(z) - g(z)$ is holomorphic, and it remains to compute this difference. In practice this is probably nontrivial, because $g(z)$ is not uniquely determined, but for functions with nice poles and principal parts, this is possible.

Such a possibility applies in your case with $f(z) = 1/(e^z + 1)$. We can justify the formula you gave in your question by using an approach based on a discussion between me and one of my friends, so I do not claim the credit for these ideas.

In order to properly handle the convergence of the infinite sum, we should first symmetrize the infinite sum you gave, so instead let $$ g(z) = \sum_{k > 0 \text{, odd}} \left( \frac{1}{k i \pi - z} - \frac{1}{k i \pi + z} \right) = -\sum_{k > 0 \text{, odd}} \frac{2z}{z^2 + k^2 \pi^2}$$

We can check that $g(z)$ is a meromorphic function whose poles and principal parts match that of $f(z) = 1/(e^z + 1)$, so it follows that $f(z)- g(z)$ is entire. It remains to compute this difference. First, notice that both $f(z)$ and $g(z)$ are $2 \pi i$ periodic. So to check the growth of $f(z), g(z)$, we need only check the behavior as $\mathrm{Re}(z) \rightarrow \pm \infty$. Notice that $f(z) \rightarrow 1,0$ as $\mathrm{Re}(z) \rightarrow -\infty, +\infty$, respectively. Thus it follows that $f(z)$ is in fact uniformly bounded away from its poles. To check $g(z)$, split the sum as $$ g(z) = \sum_{0<k<2|z|/\pi, \text{ odd}} + \sum_{k \ge 2|z|/\pi, \text{ odd}} \frac{-2z}{z^2 + k^2 \pi^2} = S_1(z) + S_2(z)$$ Now, notice that for $\mathrm{Re}(z)$ sufficiently large, \begin{align} |S_1(z)| & = \left|\frac{-2}{z} \sum_{0<k<2|z|/\pi, \text{ odd}} \frac{1}{1 + k^2 \pi^2/z^2} \right| \le \frac{2}{|z|} \frac{2|z|}{\pi} = 4,\\ |S_2(z)| & = \left| \sum_{k \ge 2|z|/\pi, \text{ odd}} \frac{-2z}{z^2 + k^2 \pi^2} \right| \\ & \le \frac{8}{3} |z| \sum_{k \ge 2|z|/\pi, \text{ odd}} \frac{1}{ \pi^2 k^2} \\ & \le \frac{8}{3} |z| \int_{-1 + 2|z|/\pi}^{\infty} \frac{1}{\pi^2 s^2} \, ds \\ & \le \frac{8}{3} \frac{|z|}{-1 + 2|z|/\pi} \le C \end{align} for $C > 0$ a constant. Thus $g(z)$ is also uniformly bounded away from its poles. Then, it follows that the difference $f(z) - g(z)$ is uniformly bounded, and being entire, then by Louiville's theorem it must be constant. Now, we compute that $$ f(0) - g(0) = \frac{1}{2} - 0 = \frac{1}{2} $$ and hence $f(z) - g(z) \equiv 1/2$.

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Awesome. Thank you very much. –  Lagerbaer May 19 '13 at 3:23

Taken literally, the claim is certainly not true. Meromorphic functions are not uniquely determined by their poles and corresponding residues. Take for example $$ \frac{1}{z} \qquad\text{and}\qquad \frac{e^z}{z}, $$ which have the same poles and same residues.

However, Mittag-Leffler's theorem as noted by Random Variable is related.


For your particular example, you can start from a well known series expansion of $\cot z$: $$\cot z = \lim_{N\to\infty} \sum_{n=-N}^N \frac{1}{z+n\pi} = \frac1z + \sum_{n=0}^\infty \frac{2z}{n^2\pi^2-z^2},$$ see for example this for a derivation.

Let's interpret your series as a symmetric sum and rewrite it by summing the terms pairwise: $$(\operatorname{term} 0 + \operatorname{term} -1) + (\operatorname{term} 1 + \operatorname{term} -2) + \cdots $$

This will give us $$ \sum_{n=-N-1}^N \frac{1}{(2n+1)i\pi - z} = -i\sum_{n=0}^N \frac{2z}{(2n+1)^2\pi^2 + z^2}, $$ which is very similar to the series for $\cot z$, but where we have thrown away half the terms. Inspired by this observation, let's compute $$ \cot z - \frac12\cot \frac z2 = \frac1z + \sum_{n=0}^\infty \frac{2z}{n^2\pi^2-z^2} - \frac1{z} - \sum_{n=0}^\infty \frac{2z}{4n^2\pi^2-z^2} = \sum_{n=0}^\infty \frac{2z}{(2n+1)^2\pi^2-z^2}. $$ Hence $$ \cot iz - \frac12\cot \frac {iz}{2} = \sum_{n=0}^\infty \frac{2z}{(2n+1)^2-(iz)^2} = \sum_{n=0}^\infty \frac{2z}{(2n+1)^2\pi^2+z^2}. $$ Consequently, your sum is \begin{align} \frac12 -i\sum_{n=0}^\infty \frac{2z}{(2n+1)^2\pi^2 + z^2} &= \frac12+\frac i2\cot \frac {iz}{2} - i\cot iz \\ &= \frac{1}{e^z+1}. \end{align}

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What part of the claim is not true? The series expansion I give above, or the general claim about meromorphic functions? –  Lagerbaer May 18 '13 at 22:04
    
The general claim. The expansion is valid, but the justification is incomplete at best. –  mrf May 18 '13 at 22:05
    
Actually, let me think about your particular expansion. Some care is needed, since the series doesn't converge absolutely. –  mrf May 18 '13 at 22:11

This seems to be some kind of application of the Residue theorem: https://en.wikipedia.org/wiki/Residue_theorem. Here, given $z_0$ at which $f$ is holomorphic, define $$g(z) = \frac{f(z)}{z - z_0}.$$ Then $g$ has poles where $f$ does, and of the same residue, but also has one extra pole, at $z_0$, whose residue we can compute via Cauchy's integral theorem to be exactly $f(z_0)$. The residue at the pole $p$ can be computed by taking $$\lim_{z\rightarrow p} zg(z) = \frac{\mbox{Res}(f,p)}{p - z_0}.$$

Now, choose a finite set of poles and your favorite point $z_0$ and a contour $\gamma$ around them so that the winding numbers are all 1 (a large box will do). Integrating around the contour will give, by the residue theorem and the above calculations:

$$f(z_0) = \frac{1}{2\pi i}\int_\gamma g(z) dz - \sum_{p \mbox{ inside } \gamma} \frac{Res(f,p)}{p - z_0}.$$

Now, the catch is to "let $\gamma$ go to infinity", to get an infinite sum. As already pointed out, there are some issues with that, so more careful estimates are needed, and these answer the question of "where did that constant come from"? I unfortunately don't have time to carry out those right now, but I will update this answer with those estimates once I have them.

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