Toronto Math Forum
MAT334-2018F => MAT334--Tests => Quiz-4 => Topic started by: Victor Ivrii on October 26, 2018, 05:55:01 PM
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Evaluate the given integral using Cauchy’s Formula or Theorem. Orientation counter-clockwise:
$$
\int_{|z|=1} \frac{\sin (z)\,dz} {z}.
$$
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\begin{equation*}
\int _{|z|\ =\ 1}\frac{sin( z)}{z}\\
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=\int _{|z|\ =\ 1} \ \frac{sin( z)}{z\ -\ 0}\\
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Set\ \zeta ( z) \ =\ sin( z)\\
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So,\ by\ Cauchy's\ formula,\\
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f( z) \ =\ \frac{1}{2\pi i}\int _{\gamma } \ \frac{\zeta ( z)}{\zeta \ -\ z}\\
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\int _{|z|\ =\ 1} \ \frac{sin( z)}{z\ -\ 0} \ =\ ( 2\pi i) \zeta ( 0) \ \\
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=\ ( 2\pi i)( sin\ 0) \ =\ 0\ \ \\
\end{equation*}
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Do not try to post solutions for many very similar problems. No double-dipping!