Cauchy Integral Theorem

The Cauchy integral theorem is a statement from the theory of functions of a complex variable .

Content

Theorem

For any function $f(z)$ ${\ displaystyle f (z)}$ analytic in some simply connected domain $A\subset \mathbb {C} ,$ ${\ displaystyle A \ subset \ mathbb {C},}$ and for any closed curve $\Gamma \subset A$ ${\ displaystyle \ Gamma \ subset A}$ fair ratio $\oint \limits _{\Gamma }\,f(z)\,dz=0$ {\ displaystyle \ oint \ limits _ {\ Gamma} \, f (z) \, dz = 0}

Proof

From the analyticity condition (Cauchy – Riemann equations) it follows that the differential form $f(z)\,dz$ ${\ displaystyle f (z) \, dz}$ closed . Let now $\Gamma$ ${\ displaystyle \ Gamma}$ - closed self-intersecting piecewise-smooth contour inside the domain of definition of the function $f(z)$ ${\ displaystyle f (z)}$ bounding area $D$ ${\ displaystyle D}$ . Then by the Stokes theorem we have:

\int \limits _{\Gamma }f(z)\,dz=\int \limits _{\partial D}f(z)\,dz=\int \limits _{D}d[f(z)\,dz]=0

{\ displaystyle \ int \ limits _ {\ Gamma} f (z) \, dz = \ int \ limits _ {\ partial D} f (z) \, dz = \ int \ limits _ {D} d [f ( z) \, dz] = 0}

Other

A limited inverse of the Cauchy theorem is Morera's theorem . A generalization of the Cauchy theorem to the case of a multidimensional complex space is the Cauchy – Poincare theorem .

Literature

Shabat B.V. Introduction to complex analysis. - M .: Science . - 1969, 577 p.