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This question involves the following homework problem:

PROBLEM

Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form

$$f(z)=\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$$

with

$$c_n= \int_0^1 f(x+iy) e^{-2\pi i n (x+iy)}dx$$

for any value $y>0$.

HINT

Show that there is an analytic function $f^*$ on a disk from which the origin is deleted such that

$$f^*(e^{2\pi i z })=f(z)$$

what is the Laurent series for $f^*$? Abbreviate $q=e^{2\pi i z }$.

MY TRY

I thought that the extension to the real line should also be analytic:

$$g(x):= \lim_{y\text{ to }0}f(x+iy)$$

and then by periodicity and the fact that $f,g$ are analytic this function is equal to its Fourier series everywhere (with possibly complex coefficients). Then by again the fact that $f,g,\sin$ and cos are analytic this is then equal to the same series using the complex series definition for $\sin$ and $\cos$. I have the idea that I am still missing something, since the hint implies a totally different approach. Thanks!

user1043065
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  • We know that $f(z)$ has a power series expansion that converges for every y>0, with coefficients given by Cauchy's formula. Now we can compose with $h(q)=\log(q)/(2 \pi i)$ such that $(f \circ h)(e^{2 \pi i z})=f(z)$ giving us $f^\star$. Now we can expand this into a power series by using Cauchy's formula for the coefficients and the fact that the logarithm is a analytic function. – Pol van Hoften Jun 02 '14 at 14:02

1 Answers1

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For the first part, compare e.g. Show this formula holds for an analytic, periodic function in each half plane or Periodic holomorphic function on the right half-plane or Entire "periodic" function and links from there. (There are even finer statements to be made once one imposes growth conditions.)

To summarize those, especially the good answers by user Martin R, periodic entire functions $f(z) = f(z+b)$ with period $b\neq 0$ are in in one-to-one correspondence with analytic functions $g$ on the punctured plane $\mathbb C \setminus \{0\}$,

and e.g. periodic functions on the upper half-plane with real period $b \neq 0$ are in one-to-one correspondence with analytic functions $g$ on the punctured unit disk $B_1(0) \setminus \{0\}$,

via defining $$g(w) = f\left(\dfrac{b}{2\pi i}\log(w)\right)$$

That is, with $g(w) = \sum_{n=-\infty}^\infty c_nw^n$, $f$ can be written as a Laurent series in $e^{\frac{2\pi i}{b} z}$, $$f(z)=\sum_{n=-\infty}^\infty c_n e^{\frac{2\pi i}{b} nz}$$.

This $g$ shall be your $f^\ast$. To get the integral formula for the coefficients, what integral would you throw on $f^\ast$ to get $c_n$? Cauchy knew. Then translate to $f$.

Torsten Schoeneberg
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