# Residue Proof of Fourier's Theorem Dirichlet Conditions

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• bolbteppa
In summary, the conversation discusses the two proofs of Fourier's theorem, one involving Dirichlet's conditions and the other using residues. The latter proof involves considering a trigonometric series with real coefficients and simplifying the partial sum to turn it into the sum of the residues of a meromorphic function. The rest of the proof involves analyzing the integration intervals and eventually arriving at Dirichlet's conditions.
bolbteppa
Whittaker (1st Edition, 1902) P.132, gives two proofs of Fourier's theorem, assuming Dirichlet's conditions. One proof is Dirichlet's proof, which involves directly summing the partial sums, is found in many books. The other proof is an absolutely stunning proof of Fourier's theorem in terms of residues, treating the partial sums as the residues of a meromorphic function and showing that, on taking the limit, we end up with Dirichlet's conditions.

(Note Later editions exclude the residue proof and prove the theorem under weaker conditions than Dirichlet's conditions in two ways, one way via the theory of summability, the other by modifying Dirichlet's proof.)

My question is about understanding the latter half of the residue proof, given here. The jist of the proof is to consider a trigonometric series with real coefficients, assume the coefficients are Fourier coefficients of a function ##f##, and then simplify the partial sum

\begin{align}
S_k(f) &= a_0 + \sum_{m=1}^k (a_m \cos(mz) + b_m \sin(mz)) \\
&= \frac{1}{2 \pi} \int_0^{2 \pi} f(t)dt + \frac{1}{\pi} \sum_{m=1}^k \int_0^{2 \pi} f(t)\cos[m(z-t)] dt \\
&= \sum_{m=-k}^k \frac{1}{2\pi} \int_0^{2 \pi} f(t)e^{im(z-t)} dt \\
&= \sum_{m=-k}^k \frac{1}{2\pi} \int_0^z f(t)e^{im(z-t)} dt + \sum_{m=-k}^k \frac{1}{2\pi} \int_z^{2 \pi} f(t)e^{im(z-t)} dt \\
&= U_k + V_k.
\end{align}
Next we try to turn ##U_k## into the sum of the residues of a meromorphic function derived from this, so try to modify it:
\begin{align}
U_k(z) &= \sum_{m=-k}^k \frac{1}{2\pi} \int_0^z f(t)e^{im(z-t)} dt \\
&= \sum_{m=-k}^k \frac{w}{2\pi w} \int_0^z f(t)e^{w(z-t)} dt |_{w = im, m \neq 0} \\
&= \sum_{m=-k}^k \frac{w}{1 + 2\pi w - 1} \int_0^z f(t)e^{w(z-t)} dt |_{w = im, m \neq 0} \\
&\to \frac{1}{1 + 2\pi w + \dots - 1} \int_0^z f(t)e^{w(z-t)} dt \\
&= \frac{1}{e^{2 \pi w} - 1} \int_0^z f(t)e^{w(z-t)} dt
\end{align}
to find
$$\phi(w) = \frac{1}{e^{2 \pi w} - 1} \int_0^z f(t)e^{w(z-t)} dt$$
so that, if ##C_k## is a circle in the ##w## plane containing ##0,i,-i,2i,-2i,\dots,ki,-ki## and no more poles, say of radius ##k+1/2##, we see
$$\frac{1}{2 \pi i} \int_{C_k} \phi(w) dw = U_k.$$
From this we integrate over the boundary explicitly via ##w = (k + 1/2)e^{i\theta}## so that ##U_k## reduces to
$$U_k = \frac{1}{2 \pi} \int_0^{2 \pi} w \phi(w) d \theta$$
and from here on we are supposed to end up with Dirichlet's conditions.

Can anybody explain the rest of the proof? Since this aspect of the proof seems to be the crux of other flawed proofs, need to make sure I get the rest of it with no hand-waving.

Thanks!

Last edited:
There is still a long way to go in the proof and too much to explain here. The next big part is to analyze the integration intervals. I suggest you specify further which part you did not get.

## What is Fourier's theorem?

Fourier's theorem is a mathematical concept that states that any periodic function can be represented as a sum of sine and cosine functions with different frequencies and amplitudes.

## What are Dirichlet conditions?

Dirichlet conditions are a set of mathematical criteria that must be met in order for Fourier's theorem to be applicable. These conditions include the function being periodic, having a finite number of discontinuities, and having a finite number of maxima and minima within each period.

## How is residue proof used in Fourier's theorem?

Residue proof is a mathematical technique used to calculate the coefficients of the sine and cosine functions in Fourier's theorem. It involves using complex analysis and the concept of residues to solve the integral equations that arise in Fourier's theorem.

## Why are Dirichlet conditions important in residue proof?

Dirichlet conditions are important in residue proof because they ensure that the integral equations used in the proof are well-defined and can be solved. Without these conditions, the proof would not be valid and Fourier's theorem could not be applied.

## What are some real-world applications of Fourier's theorem and Dirichlet conditions?

Fourier's theorem and Dirichlet conditions are used in many fields, including signal processing, image analysis, and quantum mechanics. They are also essential in solving differential equations and predicting the behavior of physical systems.

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