Complex Analysis: Use Cauchy Integral formula to evaluate...

[reverie414]

Hi, need help with this question, kinda lost.

Use the Cauchy Integral formula to evaluate foC (closed curve) f(z) dz, where C is oriented anticlockwise, for f(z) = (z + i)^2 / (z + 3 - 2i)^3 and C is the circle |z + 2| = 5.

Thanks.

 

[pka]

If g is analytic in a simply connect domain D and C is a simple closed curve in D, then if z<SUB>o</SUB> is interior to C we have:
\(\displaystyle g^{\left( n \right)} (z_0 ) = \frac{{n!}}{{2\pi i}}\int\limits_C {\frac{{g(z)}}{{\left( {z - z_0 } \right)^{n + 1} }}dz}\)
In your problem, n=2, \(\displaystyle g(z) = (z + i)^2 \quad \& \quad z_0 = - 3 + 2i.\)

Can you show that z<SUB>o</SUB> is interior to C?
Can you find the second derivative of g?