# Cauchy's integral theorem

Cauchy's integral theorem (Redirected from Cauchy integral theorem) Jump to navigation Jump to search Not to be confused with Cauchy's integral formula or Cauchy formula for repeated integration. Mathematical analysis → Complex analysis Complex analysis Complex numbers Real numberImaginary numberComplex planeComplex conjugateUnit complex number Complex functions Complex-valued functionAnalytic functionHolomorphic functionCauchy–Riemann equationsFormal power series Basic Theory Zeros and polesCauchy's integral theoremLocal primitiveCauchy's integral formulaWinding numberLaurent seriesIsolated singularityResidue theoremConformal mapSchwarz lemmaHarmonic functionLaplace's equation Geometric function theory People Augustin-Louis CauchyLeonhard EulerCarl Friedrich GaussJacques HadamardKiyoshi OkaBernhard RiemannKarl Weierstrass Mathematics portal vte In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if {displaystyle f(z)} is holomorphic in a simply connected domain Ω, then for any simply closed contour {displaystyle C} in Ω, that contour integral is zero.

{displaystyle int _{C}f(z),dz=0.} Contents 1 Statement 1.1 Fundamental theorem for complex line integrals 1.1.1 Formulation on Simply Connected Regions 1.1.2 General Formulation 1.1.3 Main Example 2 Discussion 3 Proof 4 See also 5 References 6 External links Statement Fundamental theorem for complex line integrals If f(z) is a holomorphic function on an open region U, and {displaystyle gamma } is a curve in U from {displaystyle z_{0}} to {displaystyle z_{1}} then, {displaystyle int _{gamma }f'(z),dz=f(z_{1})-f(z_{0}).} Also, when f(z) has a single-valued antiderivative in an open region U, then the path integral {textstyle int _{gamma }f'(z),dz} is path independent for all paths in U.

Formulation on Simply Connected Regions Let {displaystyle Usubseteq mathbb {C} } be a simply connected open set, and let {displaystyle f:Uto mathbb {C} } be a holomorphic function. Let {displaystyle gamma :[a,b]to U} be a smooth closed curve. Then: {displaystyle int _{gamma }f(z),dz=0.} (The condition that {displaystyle U} be simply connected means that {displaystyle U} has no "holes", or in other words, that the fundamental group of {displaystyle U} is trivial.) General Formulation Let {displaystyle Usubseteq mathbb {C} } be an open set, and let {displaystyle f:Uto mathbb {C} } be a holomorphic function. Let {displaystyle gamma :[a,b]to U} be a smooth closed curve. If {displaystyle gamma } is homotopic to a constant curve, then: {displaystyle int _{gamma }f(z),dz=0.} (Recall that a curve is homotopic to a constant curve if there exists a smooth homotopy from the curve to the constant curve. Intuitively, this means that one can shrink the curve into a point without exiting the space.) The first version is a special case of this because on a simply connected set, every closed curve is homotopic to a constant curve.

Main Example In both cases, it is important to remember that the curve {displaystyle gamma } does not surround any "holes" in the domain, or else the theorem does not apply. A famous example is the following curve: {displaystyle gamma (t)=e^{it}quad tin left[0,2pi right],} which traces out the unit circle. Here the following integral: {displaystyle int _{gamma }{frac {1}{z}},dz=2pi ineq 0,} is nonzero. The Cauchy integral theorem does not apply here since {displaystyle f(z)=1/z} is not defined at {displaystyle z=0} . Intuitively, {displaystyle gamma } surrounds a "hole" in the domain of {displaystyle f} , so {displaystyle gamma } cannot be shrunk to a point without exiting the space. Thus, the theorem does not apply.

Discussion As Édouard Goursat showed, Cauchy's integral theorem can be proven assuming only that the complex derivative {displaystyle f'(z)} exists everywhere in {displaystyle U} . This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable.

The condition that {displaystyle U} be simply connected means that {displaystyle U} has no "holes" or, in homotopy terms, that the fundamental group of {displaystyle U} is trivial; for instance, every open disk {displaystyle U_{z_{0}}={z:left|z-z_{0}right|

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