Complex Variables and Partial Differential Equations (3130005)

Question-3c ( ii )-OR

BE | Semester-3 Winter-2019 | 26-11-2019

Q3) (c ( ii ))

Here,

|z| = 1

Let,

I = \int_{C} \frac{\sin z}{{(z - \frac{π}{4})}^{3}} d z

\Rightarrow z_{0} = \frac{π}{4}

is non - analytic &

|\frac{π}{4}| = |\frac{3.14}{4}| < 1

then

\frac{π}{4} \in C

→ By comparing of,

\frac{f (z)}{{(z - z_{0})}^{n + 1}} & \frac{\sin z}{{(z - \frac{π}{4})}^{3}}

We have

f (z) = \sin z & n = 2

So,

f^{1} (z) = \cos z & f^{2} (z) = - \sin z

→ By Cauchy's integral formula,

I = \frac{2 πi}{n!} f^{n} (z_{0}) = \frac{2 πi}{2!} f^{2} (\frac{π}{4}) = πi (- \sin \frac{π}{4}) = πi (- \frac{1}{\sqrt{2}})

\Rightarrow I = - \frac{πi}{\sqrt{2}}