1) evaluate the contour integral
c.int [ e^z / ((z+1)^2 * (z-4)) ] dz
about the rectangle with verticies -3 + 4i, 3 + 4i, -3-4i, 3 - 4i,
use partial fractions and the cauchy integral formula to evaluate the integral.
2) Let C be the closed contour that runs counterclockwise around the circle | z - 2 + i | = 3
compute c.int [z - 3z^2]dz
(the z raised to the first power in this equation is barred. i.e must integrate the complex conjugate of the first z.)
cauchy integral formula f(z_0) = 1/(2*pi*i) * int [ f(z) / ( z - z_o) ] dz
The Attempt at a Solution
1. First I expanded e^z / ((z+1)^2 * (z-4)) using partial fraction decomposiion and got
e^z * ( (-1/25)/(z+1) + (-1/5)/(z+1)^2 + (1/25)/(z-4) )
brake up the integral into 3 parts
-1/25 * c.int [ e^z / (z + 1) ] dz - 1/5 * c.int [ e^z / (z + 1)^2 ]
+ 1/25 * c.int [ e^z / (z - 4) ]
applying the cauchy integral formula I get:
-1/25 *2 *pi * i * e^(-1) - 1/5 *2 *pi * i * e^(-1) + 0
the last term becomes zero because 4 lies outside the rectangle.
The only part about this that makes me uncomfortable is not using the vertecies of the rectangle at all in the computation of the curve. Did I need to parameterize the rectangle? Thanks
2) for the integral c.int [z - 3z^2]dz i parametrized the circle | z - 2 + i | = 3 as
z(t) = 2 + i + 3e^(it)
z'(t) = i3e^(it)
then evaluated c.int [ f(z(t))z'(t)dt ]
could someone please verify my methods.