Cauchy Integral Formula

\begin{array}{l} C a u c h y I n t e g r a l F o r m u l a \begin{matrix}  \end{matrix} \\ T h e o r e m : L e t f b e a n a l y t i c e v e r y w h e r e i n s i d e a n d o n c l o s e d c o n t o u r C, \\ t a k e n i n t h e p o s i t i v e s e n s e . \\ I f z_{0} i s a n y p o i n t i n t e r i o r t o C, t h e n \\ \int_{C}^{} \frac{f (z)}{z - z_{0}} d z = 2 π i f (z_{0}) \end{array}

\begin{array}{l} E x a m p l e : \\ \int_{| z | = 3}^{} \frac{z^{2} + 2 z}{z - 2} d z \\ S o l u t i o n : \\ L e t C : | z | = 3. S i n c e t h e f u n c t i o n f (z) = z^{2} + 2 z, i s a n a l y t i c \\ w i t h i n a n d o n C a n d s i n c e \end{array}

\begin{array}{l} t h e p o i n t z_{0} = 2 i s i n C . \\ T h e n b y C a u c h y i n t e g r a l f o r m u l a \\ \int_{| z | = 3}^{} \frac{z^{2} + 2 z}{z - 2} d z = \int_{| z | = 3}^{} \frac{f (z)}{z - z_{0}} d z \\ 2 π i f (2), (f (z) = z^{2} + 2 z) \\ 2 π i (4 + 4) \\ 16 π i . \end{array}