Cauchy's Theorem and Integral Formula

The single fact from which nearly all of complex analysis follows is that the integral of a holomorphic function around a closed loop is zero. From it come the Cauchy integral formula — a function's values inside a contour are fixed by its values on the contour — and a cascade of rigidity theorems with no real- analysis analogue. This page builds on holomorphic.md and is the foundation for residues.

Contour integrals

For a piecewise-smooth curve $\gamma$ and continuous $f$, the contour integral is $${\int }_{\gamma }f(z)dz={\int }_a^{b}f(\gamma (t)){\gamma }^{\prime }(t)dt.$$ It is linear, reverses sign under reversal of $\gamma$, and is bounded by the ML inequality $\left|{\int }_{\gamma }fdz\right|\le ({\max }_{\gamma }|f|)\cdot \mathrm{length}(\gamma )$. The basic computation, for a counterclockwise circle of radius $r$ about $a$: $${\oint }_{|z-a|=r}(z-a{)}^{n}dz=\{\begin{array}{ll}2\pi i& n=-1\\ 0& n\ne -1\end{array}(n\in \mathbb{Z}).$$ The lone surviving power $n=-1$ is the seed of the entire residue calculus.

Cauchy's theorem

Cauchy–Goursat theorem. If $f$ is holomorphic on a simply connected domain $U$, then ${\oint }_{\gamma }fdz=0$ for every closed contour $\gamma$ in $U$.

Why it is Green's theorem plus CR. Writing $f=u+iv$ and $dz=dx+idy$, $${\oint }_{\gamma }fdz=\oint (udx-vdy)+i\oint (vdx+udy),$$ and Green's theorem converts each loop integral to a double integral of $(-v_x-u_y)$ and $(u_x-v_y)$ — both zero by the Cauchy–Riemann equations. So Cauchy's theorem is the planar Stokes theorem specialized to a holomorphic integrand. (Goursat's achievement was removing the continuity-of-derivative hypothesis Green's theorem needs.) On a multiply connected domain the integral need not vanish — it depends only on the homotopy class of $\gamma$, a topological ($H^1$) statement that foreshadows residues and matches de-rham.md.

Winding numbers and the homology form

The simply connected statement is a special case of a sharper one that says exactly when ${\oint }_{\gamma }f=0$ on an arbitrary domain. The measuring device is the winding number (index) of a closed curve about a point $a\notin \gamma$: $$n(\gamma ,a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{dz}{z-a}\in \mathbb{Z},$$ the net number of counterclockwise turns $\gamma$ makes around $a$ (an integer by the $n=-1$ computation above, and locally constant in $a$). A cycle $\gamma$ is null-homologous in $U$ if $n(\gamma ,a)=0$ for every $a\notin U$ — it does not enclose any point outside the domain.

Cauchy's theorem (homology form). If $f$ is holomorphic on a domain $U$ and $\gamma$ is a cycle that is null-homologous in $U$, then ${\oint }_{\gamma }fdz=0$; more generally, for $z_0\in U\setminus \gamma$, $$n(\gamma ,z_0)f(z_0)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-z_0}dz.$$

This subsumes both Cauchy's theorem and the integral formula (take $\gamma$ a simple loop, $n=1$), holds on any domain (no simple-connectivity hypothesis), and makes precise that the integral sees only the homology class of $\gamma$ in $U$. A domain is simply connected iff every cycle in it is null-homologous — the topological characterization that recovers the first statement. The winding number is also the engine of the argument principle.

Path independence and primitives

Cauchy's theorem is equivalent to path independence: on a simply connected domain ${\int }_{\gamma }fdz$ depends only on the endpoints, so $f$ has a primitive $F$ with $F^{\prime }=f$ (define $F(z)={\int }_{z_0}^{z}f$). This is the complex fundamental theorem of calculus; it fails on non-simply-connected domains exactly when $f$ has a nonzero residue (e.g. $1/z$ on $\mathbb{C}\setminus \{0\}$, whose "primitive" $\log z$ is multivalued — holomorphic.md).

The Cauchy integral formula

The keystone: deform any loop around $z_0$ to a small circle and use the $n=-1$ computation.

Cauchy integral formula. If $f$ is holomorphic inside and on a positively oriented simple closed contour $\gamma$, then for $z_0$ inside, $$f(z_0)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-z_0}dz,$$ and differentiating under the integral, $$f^{(n)}(z_0)=\frac{n!}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{(z-z_0{)}^{n+1}}dz.$$

The values of $f$ on the boundary determine $f$ and all its derivatives inside — a holomorphic function is fantastically over-determined.

The rigidity corollaries

The formula immediately yields the theorems that give complex analysis its distinctive flavor:

• Holomorphic ⇒ infinitely differentiable ⇒ analytic (the formula for $f^{(n)}$ exists for all $n$; the Taylor series converges — see holomorphic.md).
• Liouville's theorem. A bounded entire function is constant. (From the $n=1$ formula and the ML bound as the radius $\to \infty$.) Immediate corollary: the fundamental theorem of algebra — every non-constant polynomial has a root.
• Maximum modulus principle. A non-constant holomorphic function has no interior maximum of $|f|$; extrema live on the boundary. (The mean-value property: $f(z_0)$ is the average of $f$ over any circle about it.)
• Morera's theorem (converse to Cauchy): if ${\oint }_{\gamma }f=0$ for all loops, $f$ is holomorphic — the tool for proving limits/integrals of holomorphic functions are holomorphic.
• Cauchy estimates and Schwarz's lemma, bounding derivatives and maps of the disc.

References

• Ahlfors, Complex Analysis, Ch. 4.
• Stein & Shakarchi, Complex Analysis, Ch. 2.
• Conway, Functions of One Complex Variable, Ch. IV.