variant of Cauchy integral formula

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

$f(z)={\displaystyle \frac{1}{2i\pi }}{\displaystyle {\oint }_C}{\displaystyle \frac{f(t)}{t-z}}𝑑t.$(1)

Proof.  Let $\epsilon$ be any positive number.  Denote by $C_r$ the circles with radius $r$ and centered in $z$.  We have

$${\oint }_C\frac{f(t)}{t-z}𝑑t={\oint }_C\frac{f(z)+(f(t)-f(z))}{t-z}𝑑t=\underset{I}{\underbrace{{\oint }_C\frac{f(z)}{t-z}𝑑t}}+\underset{J}{\underbrace{{\oint }_C\frac{f(t)-f(z)}{t-z}𝑑t}}.$$

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

$$I=f(z){\oint }_C\frac{dt}{t-z}=\mathrm{\hspace{0.33em}2}i\pi f(z).$$

If  ,  we have

$$J={\oint }_{C_r}\frac{f(t)-f(z)}{t-z}𝑑t.$$

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

when    i.e. when

(2)

If (2) is in , we obtain first

whence, by the estimation theorem of integral,

and lastly

(3)

This result implies (1).