Complex Analysis
Integral Formula
Draft for Information Only ContentCauchy's Theorem and Integral Formula source/reference: Cauchy's Theorem and Integral FormulaCauchy's TheoremTheorem (Cauchy's Theorem for Simply Connected Domains)Let 𝐷 be a simply connected domain in ℂ, and let 𝑓 be analytic in 𝐷. Let 𝛾:[𝑎,𝑏]→𝐷 be a piecewise smooth, closed curve in 𝐷 (i.e. 𝛾(𝑏)=𝛾(𝑎)). Then
Example𝑓(𝑧)=ℯ^{(𝑧3)} is analytic in ℂ, and ℂ is simply connected. Therefore, Proof ideaSince 𝐷 has no holes, 𝛾 can be deformed continuously to a point in 𝐷. Show that the integral does not change along the way by using the Cauchy Theorem in a disk.
A First conclusionCorollaryLet 𝛾_{1} and 𝛾_{2} be two simple closed curves (i.e. neither of the curves intersects itself), oriented counterclockwise, where
𝛾_{2} is inside 𝛾_{1}. If 𝑓 is analytic in a domain 𝐷 that contains both curves as well as the region between them, then
Proof ideaA neat trick: Form a "joint curve" 𝛾 as in the picture below. As 𝑓 is analytic in a simply connected region, containing 𝛾, thus have
Examples
The Cauchy Integral FormulaTheorem (Cauchy Integral Formula)Let 𝐷 be a simply connected domain, bounded by a piecewise smooth curve 𝛾, and let 𝑓 be analytic in a set 𝑈 that contains the closure of 𝐷 (i.e. 𝐷 and 𝛾). Then
The Proof of the Cauchy Integral FormulaThe proof of the Cauchy Integral Formula goes as follows:
ExamplesLet 𝑓(𝑤)=
Analyticity of the DerivativeHere is an amazing consequence of the Cauchy Integral Formula: TheoremIf 𝑓 is analytic in an open set 𝑈, then 𝑓′ is also analytic in 𝑈.
Idea of Proof
The Cauchy Integral Formula for DerivativesRepeated application of the previous theorem shows that an analytic function has infinitely many derivatives! Continuing along the same lines as the previous proof yields the following extension of the Cauchy Integral Formula: Theorem (Cauchy Integral Formula for Derivatives)Let 𝐷 be a simply connected domain, bounded by a piecewise smooth curve 𝛾, and let 𝑓 be analytic in a set 𝑈 that contains the closure of 𝐷 (i.e. 𝐷 and 𝛾). Then
where, 𝑓^{(𝑘)} denotes the 𝑘th derivative of 𝑓. ExamplesLet
Summary
Cauchy's EstimateTheorem (Cauchy's Extimate)Suppose that 𝑓 is analytic in an open set that contains
ProofBy the Cauchy Integral Formula, having that
Liouville's TheoremTheorem (Liouville)Let 𝑓 be analytic in the complex plane (thus 𝑓 is an entire function). If 𝑓 is bounded then 𝑓 must be constant.
ProofSuppose that 𝑓(𝑧)≤𝑚 for all 𝑧∈ℂ. Pick 𝑧_{0}∈ℂ. Since ℂ contains
ExampleExampleSuppose that 𝑓 is an entire function, 𝑓=𝑢+𝑖𝑣, and suppose that 𝑢(𝑧)≤0 for all 𝑧∈ℂ. Then 𝑓 must be constant.
ProofConsider the function 𝑔(𝑧)=ℯ^{Re 𝑓(𝑧)}. Then 𝑔 is an entire function as well. Furthermore,
Use Liouville to Prove Fundamentatl Theorem of AlgebraTheorem (Fundamental Theorem of Algebra)
ProofSuppose to the contrary that there exists a polynomial 𝑝 as in the theorem that has no zeros. Then 𝑓(𝑧)=
Factoring of PolynomialsConsequence of the Fundamental Theorem of Algebra: Polynomials can be factored in ℂ
Expample𝑝(𝑥)=𝑥^{2}+1 has no zeros in ℝ, thus cannot be factored in ℝ. However, in ℂ, 𝑝(𝑧)=𝑧^{2}+1 has two zeros 𝑖 and −𝑖 and thus factors as 𝑝(𝑧)=(𝑧−𝑖)(𝑧+𝑖)
The Maximum PrincipleAnother consequence of the Cauchy Integral Formula is the following powerful result. Theorem (Maximum Principle)Let 𝑓 be analytic in a domain 𝐷 and suppose there exists a point 𝑧_{0}∈𝐷 such that 𝑓(𝑧)≤𝑓(𝑧_{0}) for all 𝑧∈𝐷. Then 𝑓 is constant in 𝐷.
ConsequenceIf 𝐷⊂ℂ is a bounded domain, and if 𝑓:
ExampleLet 𝑓(𝑧)=𝑧^{2}2𝑧. What is max𝑓(𝑧) on the square 𝑄={𝑧=𝑥+𝑖𝑦:0≤𝑥,𝑦≤1}? since 𝑓 is analytic inside 𝑄 and continuous on 𝑄, the maximum of 𝑓 occurs on ∂𝑄.
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