variant of Cauchy integral formula

Theorem. Let $f(z)$ be holomorphic in a domain $A$ of $\mathbb{C}$ . If $C$ is a closed contour not intersecting itself which with its domain is contained in $A$ and if $z$ is an arbitrary point inside $C$ , then

$\displaystyle f(z)\;=\;\frac{1}{2i\pi}\oint_{C}\frac{f(t)}{t\!-\!z}\,dt.$

(1)

Proof. Let $\varepsilon$ be any positive number. Denote by $C_{r}$ the circles with radius $r$ and centered in $z$ . We have

$\oint_{C}\frac{f(t)}{t\!-\!z}\,dt\;=\;\oint_{C}\frac{f(z)\!+\!(f(t)\!-\!f(z))}% {t\!-\!z}\,dt\;=\;\underbrace{\oint_{C}\frac{f(z)}{t\!-\!z}\,dt}_{I}+% \underbrace{\oint_{C}\frac{f(t)\!-\!f(z)}{t\!-\!z}\,dt}_{J}.$

According to the corollary of Cauchy integral theorem and its example, we may write

$I\;=\;f(z)\oint_{C}\frac{dt}{t\!-\!z}\;=\;2i\pi f(z).$

If $0<r<\mbox{ some }r_{0}$ , we have

$J\;=\;\oint_{C_{r}}\frac{f(t)\!-\!f(z)}{t\!-\!z}\,dt.$

The continuity of $f$ in the point $z$ implies, that

$|f(t)\!-\!f(z)|<\varepsilon$

when $0<|t\!-\!z|<\mbox{ some }\delta_{\varepsilon}$ i.e. when

$\displaystyle t\in C_{r}\,\mbox{ and }\,0<r<\mbox{ some }r_{1}.$

(2)

If (2) is in , we obtain first

$\left|\frac{f(t)\!-\!f(z)}{t\!-\!z}\right|\;=\;\frac{|f(t)\!-\!f(z)|}{|t\!-\!z% |}\;=\;\frac{|f(t)\!-\!f(z)|}{r}\;<\;\frac{\varepsilon}{r},$

whence, by the estimation theorem of integral,

$|J|\;\leqq\;\frac{\varepsilon}{r}\cdot 2\pi r\;=\;2\pi\varepsilon\quad\mbox{% for}\quad 0<r<\min\{r_{0},\,r_{1}\},$

and lastly

$\displaystyle\left|\frac{1}{2i\pi}\oint_{C}\frac{f(t)}{t\!-\!z}\,dt-f(z)\right% |\;=\;\left|\frac{1}{2i\pi}J\right|\;\leqq\;\frac{1}{2\pi}\cdot 2\pi% \varepsilon\;=\;\varepsilon\quad\mbox{when }0<r<\min\{r_{0},\,r_{1}\}.$

(3)

This result implies (1).

Title	variant of Cauchy integral formula
Canonical name	VariantOfCauchyIntegralFormula
Date of creation	2013-03-22 18:54:15
Last modified on	2013-03-22 18:54:15
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Theorem
Classification	msc 30E20
Synonym	Cauchy integral formula
Related topic	CauchyIntegralFormula
Related topic	CorollaryOfCauchyIntegralTheorem
Related topic	ExampleOfFindingTheGeneratingFunction
Related topic	GeneratingFunctionOfLaguerrePolynomials
Related topic	GeneratingFunctionOfHermitePolynomials