Cauchy Integral Formula
Published On :2021-06-27 01:13:00
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Cauchy;Integral;Formula;begin{array}{*{20}{l}}\
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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{{fleft( z right)}}{{z - {z_0}}}dz = 2pi i;fleft( {{z_0}} right)
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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;fleft( z right) = {z^2} + 2z;,;is;analytic;\within;and;on;C;and;since
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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{{fleft( z right)}}{{z - {z_0}}}dz\\
2pi i;fleft( 2 right);;;;;;;;;;,;;;;;left( {fleft( z right) = {z^2} + 2z} right)\\
2pi i;left( {4 + 4} right)\\
16pi i.\
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