holomorphic function associated with continuous function

Theorem.  If $f(z)$ is continuous on a (finite) contour $\gamma$ of the complex plane, then the contour integral

$g(z)=:{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(t)}{t-z}}dt,$ (1)

defines a function  $z\mapsto g(z)$  which is holomorphic in any domain $D$ not containing points of $\gamma$.  Moreover, the derivative has the expression

$g^{\prime }(z)={\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(t)}{{(t-z)}^2}}𝑑t.$ (2)

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

$$\underset{\mathrm{\Delta }z\to 0}{lim}\frac{g(z+\mathrm{\Delta }z)-g(z)}{\mathrm{\Delta }z}.$$

Let  $z_1=:z+\mathrm{\Delta }z\notin \gamma$,  $\mathrm{\Delta }z\ne 0$.  We may write first

$$\frac{g(z_1)-g(z)}{z_1-z}=\frac{1}{\mathrm{\Delta }z}{\int }_{\gamma }f(t)\left[\frac{1}{t-z_1}-\frac{1}{t-z}\right]𝑑t={\int }_{\gamma }\frac{f(t)}{(t-z_1)(t-z)}𝑑t,$$

whence

$$E=:\frac{g(z_1)-g(z)}{z_1-z}-{\int }_{\gamma }\frac{f(t)}{{(t-z)}^2}=\mathrm{\Delta }z\cdot {\int }_{\gamma }\frac{f(t)}{(t-z_1){(t-z)}^2}dt.$$

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

As well, we have a positive constant $d$ such that

$$|t-z|\geqq d\mathit{\hspace{1em}}\forall t\in \gamma .$$

When we choose  ,  it follows that

$$|t-z_1|=|(t-z)-\mathrm{\Delta }z|\geqq |t-z|-|\mathrm{\Delta }z|>d-\frac{d}{2}=\frac{d}{2}.$$

Consequently,

and, by the estimating theorem of contour integral,

where $k$ is the length of the contour.  The last expression tends to zero as  $\mathrm{\Delta }z\to 0$.  This settles the proof.

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

$g^{(n)}(z)=n!{\displaystyle {\int }_{\gamma }}{\displaystyle \frac{f(t)}{{(t-z)}^{n+1}}}dt\mathit{\hspace{1em}\hspace{1em}}(n=\mathrm{\hspace{0.33em}0},\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$ (3)

Remark 2.  The contour $\gamma$ may be .  If it especially is a circle, then (1) defines a holomorphic function inside $\gamma$ 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