MAT334-2018F > Quiz-4

Q4 TUT 5101

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

Jeffery Mcbride:

\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*}

Victor Ivrii:
Do not try to post solutions for many very similar problems. No double-dipping!

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