Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz4 => Topic started by: Victor Ivrii on October 26, 2018, 05:55:01 PM

Evaluate the given integral using Cauchy’s Formula or Theorem. Orientation counterclockwise:
$$
\int_{z=1} \frac{\sin (z)\,dz} {z}.
$$

\begin{equation*}
\int _{z\ =\ 1}\frac{sin( z)}{z}\\
\\
=\int _{z\ =\ 1} \ \frac{sin( z)}{z\ \ 0}\\
\\
Set\ \zeta ( z) \ =\ sin( z)\\
\\
So,\ by\ Cauchy's\ formula,\\
\\
f( z) \ =\ \frac{1}{2\pi i}\int _{\gamma } \ \frac{\zeta ( z)}{\zeta \ \ z}\\
\\
\int _{z\ =\ 1} \ \frac{sin( z)}{z\ \ 0} \ =\ ( 2\pi i) \zeta ( 0) \ \\
\\
=\ ( 2\pi i)( sin\ 0) \ =\ 0\ \ \\
\end{equation*}

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