Cauchy's Integral Formula

Subjects: Complex Analysis
Links: Homotopy Cauchy's Theorem, Homology Cauchy's Theorem, Analytic Functions

Def: Let $γ$ be a closed curve in $C$ and $z_{0} \in C$ be a point not on $γ$ . Then, the index of $γ$ with respect to $z_{0}$ (also called the winding number of $γ$ with respect to $z_{0}$ ) is defined to be

n (γ; z_{0}) = \frac{1}{2 π i} \oint_{γ} \frac{d z}{z - z_{0}}

we say that $γ$ ************winds around $z_{0}$ , $n (γ, z_{0})$ times.

Prop:

The circle $γ (t) = z_{0} + r e^{i t}$ , with $r > 0$ is the radius and the parameter $0 \leq t \leq 2 π n$ , has an index of $n$ with respect to $z_{0}$ ; while the circle $γ^{-} (t) = z_{0} + e^{- i t}$ , where $0 \leq t \leq 2 π n$ , has an index of $- n$
If $z_{0}$ does not lie on either $\tilde{γ}$ or $γ$ , and if $\tilde{γ}$ and $γ$ are homotopic in $C ∖ {z_{0}}$ , then

n (\tilde{γ}; z_{0}) = n (γ; z_{0})

Th: Let $γ : [a, b] \to C$ be a piecewise smooth closed curve and $z_{0}$ a point not on $γ$ ; then $I (γ; z_{0})$ is an integer.

Let $γ : [a, b] \to C$ be a piecewise smooth closed curve.

If $z_{0} \in C$ and $δ > 0$ such that $γ \subseteq B_{δ} (z_{0})$ , then $n (γ; w) = 0$ for all $w \notin B_{δ} (z_{0})$
If $z, w \in C$ belong to the same connected component of $C ∖ γ$ , then $n (γ; z) = n (γ; w)$
If $z \in C$ belongs to the unbounded connected component of $C ∖ γ$ , then $n (γ; z) = 0$

Cauchy’s Integral Formula (for disks)

Let $z_{0} \in C$ , $r > 0$ and $f : B_{r} (z_{0}) \to C$ . If $f$ is analytic on $B_{r} (z_{0})$ and $γ \subseteq B_{r} (z_{0})$ a be a piecewise smooth closed curve, then

f (z_{0}) \cdot n (γ; z_{0}) = \frac{1}{2 π i} \oint_{γ} \frac{f (z)}{z - z_{0}} d z

Homotopy Cauchy's Integral Formula

Let $f$ be analytic on a region $A$ . Let $γ$ be closed curve in $A$ that is homotopic to a point, let $z_{0} \in A$ be a point not on $γ$ . Then

f (z_{0}) \cdot n (γ; z_{0}) = \frac{1}{2 π i} \oint_{γ} \frac{f (z)}{z - z_{0}} d z

If we say that $γ$ is a simple closed curve, and $z_{0}$ is “inside” of $γ$ . Then $I (γ; z_{0}) = 1$ , so the formula is

f (z_{0}) = \frac{1}{2 π i} \oint_{γ} \frac{f (z)}{z - z_{0}} d z

There’s a more powerful version where $f$ is continuous on $γ$ , and holomorphic on the “inside” of $γ$ , and the formula is still valid. The inside of the curve is defined using the Jordan Curve Theorem

Homology Cauchy Integral Formula

Let $f : Ω \subseteq C \to C$ analytic in the region $Ω$ and $γ \subseteq Ω$ a cycle such that $γ \sim 0 (\mod Ω)$ . If $z_{0} \in Ω ∖ γ$ , then

n (γ; z_{0}) f (z_{0}) = \frac{1}{2 π i} \oint_{γ} \frac{f (z)}{z - z_{0}}

Let the function $f$ be holomorphic in the closure of a compact domain $D$ that is bounded by a finite number of continuous curves. Then the function f at any point $z \in D$ may be represented as

f (z) = \frac{1}{2 π i} \int_{\partial D} \frac{f (ζ)}{ζ - z} d ζ

where $\partial D$ is the oriented boundary of $D$ .

Generalized Cauchy’s Integral Formula

Let $f$ be a complex function with continuous Wirtinger Derivatives in a closure of a compact region $D$ bounded by by a finite number of piecewise smooth curves. Then we have

f (z) = \frac{1}{2 π i} \oint_{\partial D} \frac{f (ζ)}{ζ - z} d ζ - \frac{1}{π} \int_{A} \frac{\partial f / \partial \overset{―}{ζ}}{ζ - z} d A (ζ, \overset{―}{ζ})

We are using the Green's Theorem analogue but for contour integrals