holomorphic function associated with continuous function

TheoremMathworldPlanetmath.  If f(z) is continuousMathworldPlanetmathPlanetmath on a (finite) contour γ of the complex plane, then the contour integral

g(z)=:γf(t)t-zdt, (1)

defines a functionzg(z)  which is holomorphic in any domain D not containing points of γ.  Moreover, the derivativePlanetmathPlanetmath has the expression

g(z)=γf(t)(t-z)2𝑑t. (2)

Proof.  The right hand side of (2) is defined since its integrand is continuous.  On has to show that it equals


Let  z1=:z+Δzγ,  Δz0.  We may write first




Because f is continuous in the compact set γ, there is a positive constant M such that


As well, we have a positive constant d such that


When we choose  |Δz|<d2,  it follows that




and, by the estimating theorem of contour integral,


where k is the length of the contour.  The last expression tends to zero as  Δz0.  This settles the proof.

Remark 1.  By inductionMathworldPlanetmath, one can prove the following generalisation of (2):

g(n)(z)=n!γf(t)(t-z)n+1dt  (n= 0, 1, 2,) (3)

Remark 2.  The contour γ may be .  If it especially is a circle, then (1) defines a holomorphic function inside γ and another outside it.

Title holomorphic function associated with continuous function
Canonical name HolomorphicFunctionAssociatedWithContinuousFunction
Date of creation 2013-03-22 19:14:29
Last modified on 2013-03-22 19:14:29
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 30E20
Classification msc 30D20
Related topic DifferentiationUnderIntegralSign
Related topic CauchyIntegralFormula