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 inner domain is contained in $A$ and if $z$ is an arbitrary point inside $C$ , then

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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 $0 < r < \mbox{ some } r_0$ , 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

and     some

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when

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