# 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,  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 (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

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 (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