variant of Cauchy integral formula

Theorem.  Let f(z) be holomorphic in a domain A of .  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

f(z)=12iπCf(t)t-z𝑑t. (1)

Proof.  Let ε be any positive number.  Denote by Cr the circles with radius r and centered in z.  We have


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

I=f(z)Cdtt-z= 2iπf(z).

If  0<r< some r0,  we have


The continuity of f in the point z implies, that


when  0<|t-z|< some δε  i.e. when

tCr and  0<r< some r1. (2)

If (2) is in , we obtain first


whence, by the estimation theorem of integral,

|J|εr2πr= 2πεfor0<r<min{r0,r1},

and lastly

|12iπCf(t)t-z𝑑t-f(z)|=|12iπJ|12π2πε=εwhen 0<r<min{r0,r1}. (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 formulaPlanetmathPlanetmath
Related topic CauchyIntegralFormula
Related topic CorollaryOfCauchyIntegralTheorem
Related topic ExampleOfFindingTheGeneratingFunction
Related topic GeneratingFunctionOfLaguerrePolynomials
Related topic GeneratingFunctionOfHermitePolynomials