Reference no: EM132312145
Assignment -
Question 1 -
a. Draw sketches of the following regions in the complex plane:
i. |z-2i| = 4
ii. Im(z2) = 1, - π/2 < arg(z) < π/2
b. Use the definitions of cosh(z) and sinh(z) to show
cosh(π/2 i - z) = sinh(z)/i
c. Find all solutions for z of cos z = 2i
d. Find all values for the following expressions. Also identify the principal value. Put all values in the form a + ib
i. (-i)1/4
ii. (cosi)i
iii. log(1-i)
Question 2 -
a. Determine, if possible, the following limits
i. limz→0(z-z-)2/|z|2
ii. limz→0(ez-1)/sinz [Use Taylor series, or any other available method].
b. Let u(x, y) = xy + cosh x sin y
i. Show u(x, y) is harmonic.
ii. Find a harmonic conjugate, v(x, y) of u(x, y).
iii. Find a function, f(z) such that Re(f(z)) = u(x, y).
iv. Calculate the derivative, f'(z).
Question 3 -
a. Let C be the curve given in the following diagram:
![1801_figure.png](https://secure.expertsmind.com/CMSImages/1801_figure.png)
[C is a straight from 0 to 1, then goes from 1 to i along the circle centred at the origin with radius 1. It then follows a straight line back to the origin.]
Calculate
i.
C z2dx
ii.
C z-^2 dz
b. Find
|z|=2 (2z3+iz)/(z+i)3 dz [Note: the counter is |z| = 2]
c. Find
|z|=1/2 sinhz/(z4-1) dz [Note: the counter is |z| = 1/2]
Question 4 -
a. Let f(z) = 1/(z2-1)
i. Find the Laurent series for which f(z) converges for 0 < |z+1| < 2.
ii. Find the Taylor series of f(z) about z = i and state the region where the series converges.
b. Find the Laurent series for z3 cos(1/z) in powers of z. What is the nature of the singularity at z = 0?
Question 5 -
a. Evaluate the following integrals [note: the contour is |z| = 2]:
i.
|z|=2 z3 cos(1/z)dz
ii.
|z|=2z/(z2+z+1)dz
b. Show 0∫π 8dθ/(5+2cosθ) = 1/i
|z|=1 4dz/(z2+5z+1) and hence evaluate the integral.
c. Evaluate, using residue calculus:
-∞∫∞ dx/((x2+4)(x2+1))