Cauchy Integral Formula

 

\[\begin{array}{l} Cauchy\;Integral\;Formula\;\begin{array}{*{20}{l}}\\ \;\\ \; \end{array}\\\\ Theorem:\;Let\;f\;be\;analytic\;everywhere\;inside\;and\;on\;closed\;contour\;C,\;\\taken\;in\;the\;positive\;sense.\;\\\\ If\;{z_0}\;is\;any\;point\;interior\;to\;C,\;then\\\\ \mathop \smallint \nolimits_C^\; \frac{{f\left( z \right)}}{{z - {z_0}}}dz = 2\pi i\;f\left( {{z_0}} \right) \end{array}\]

\[\begin{array}{l} Example:\\\\ \mathop \smallint \nolimits_{\left| z \right| = 3}^\; \frac{{{z^2} + 2z}}{{z - 2}}dz\\\\ Solution:\\\\ Let\;C\;:\;\left| z \right| = 3.\;Since\;the\;function\;f\left( z \right) = {z^2} + 2z\;,\;is\;analytic\;\\within\;and\;on\;C\;and\;since \end{array}\]

\[\begin{array}{l} the\;point\;{z_0} = 2\;\;\;is\;in\;C.\\\\ Then\;by\;Cauchy\;integral\;formula\\\\ \mathop \smallint \nolimits_{\left| z \right| = 3}^\; \frac{{{z^2} + 2z}}{{z - 2}}dz = \mathop \smallint \nolimits_{\left| z \right| = 3}^\; \frac{{f\left( z \right)}}{{z - {z_0}}}dz\\\\ 2\pi i\;f\left( 2 \right)\;\;\;\;\;\;\;\;\;\;,\;\;\;\;\;\left( {f\left( z \right) = {z^2} + 2z} \right)\\\\ 2\pi i\;\left( {4 + 4} \right)\\\\ 16\pi i.\\ \end{array}\]