nLab Goursat theorem

Idea

In complex analysis, the Goursat theorem is the extension (due to Édouard Goursat) of the Cauchy integral theorem from continuously differentiable functions (for which Augustin Cauchy had proved it) to differentiable functions (which requires a harder and more technical argument). Given Cauchy’s other work, the immediate corollary is that every differentiable function is in fact analytic; see holomorphic function. This corollary may also be viewed as the Goursat theorem; everything else in the basic theory of holomorphic functions is due to Cauchy, while Goursat's contribution was to remove the hypothesis that the derivative of such a function must be continuous. On the other hand, the Goursat theorem may be seen as not a new theorem at all; from this perspective, Cauchy had always intended to prove his integral theorem for all complex-differentiable functions, and Goursat simply filled in a gap in Cauchy's proof. But as Goursat was born the year after Cauchy died, that was a long-standing gap!

Statements

Let $U$ be an open subset of the complex plane $\mathbb{C}$ and let $f\colon U \to \mathbb{C}$ be a complex-differentiable function; that is,

$$f'(\zeta) \coloneqq \lim_{z \to \zeta} \frac{f(z) - f(\zeta)}{z - \zeta}$$

exists for every point $\zeta$ in $U$. Then any of the following conclusions may be thought of as the Goursat theorem:

Theorem

The function $f'\colon U \to \mathbb{C}$ is continuous. That is, $f$ is continuously differentiable.

Theorem

Given any point $\zeta$ in $U$ and any rectangle (or triangle) $C$ in $U$ whose inside contains $\zeta$ and is contained in $U$, the contour integral

$$\oint_C f(z) \,\mathrm{d}z = 0 .$$

That is, the Cauchy integral theorem holds for $f$ on rectangles (or triangles).

Theorem

Given any point $\zeta$ in $U$ and any Jordan curve $C$ in $U$ whose inside contains $\zeta$ and is contained in $U$, the contour integral

$$\oint_C f(z) \,\mathrm{d}z = 0 .$$

That is, the Cauchy integral theorem holds for $f$.

Theorem

Given any point $\zeta$ in $U$ and any Jordan curve $C$ in $U$ whose inside contains $\zeta$ and is contained in $U$, the contour integral

$$\oint_C \frac{f(z)}{z - \zeta} \,\mathrm{d}z = 2\pi\mathrm{i}f(\zeta) .$$

That is, the Cauchy integral formula holds for $f$.

Theorem

Given any point $\zeta$ in $U$, there is an infinite sequence $(c_0, c_1, \ldots)$ such that

$$f(z) = \sum_i c_i (z - \zeta)^i$$

on some neighbourhood of $\zeta$ contained in $U$. That is, the function $f$ is analytic.

Cauchy had established that the conclusion of each of these theorems implied the conclusion of the next (except that he did not bother with Theorem ), and it's immediate that the last implies the first, so from the perspective of Goursat's time, these theorems are all equivalent. Goursat actually introduced his proof for Theorem , which is of interest only because Goursat's proof naturally gives it. Since Goursat was the first to call attention to it, one might call Theorem specifically Goursat's Theorem, even though it is more of a lemma (for Theorem ) than a theorem in its own right. (One might call Theorem ‘Goursat's Lemma’, but that is already used for an unrelated result about pullbacks in group theory.)

Proof

Here's a version for Theorem : https://planetmath.org/proofofgoursatstheorem