Fall 2011-MTH5130-Theory of a Complex Variable-Homework 5
1. Given n > 1, nd out how many non-zero complex numbers z exists such that one of the
nth degree roots of z is z . Plot all these complex numbers on a plane. Demonstrate
your answer if n = 5.
2. L
Fall 2011-MTH5130-Theory of a Complex Variable-Homework 6
1. Evaluate the integral
|z |zdz,
C
where C is the closed contour consisting of the upper semicircle |z | = 1 and the
segment 1 x 1, y = 0.
2. In the two problems below the branch of the many-value
Fall 2011-MTH5130-Theory of a Complex Variable-Homework 4
1. A function u(x, y ) which in some domain possesses continuous partial derivatives up to
second order inclusive and satises Laplaces equation
u uxx + uyy = 0
is said to be harmonic function. Two
Fall 2011-MTH5130-Theory of a Complex Variable-Homework 1
1. Perform the operations indicated:
2
; (1 + i 3)3 ;
1 3i
2. Find the modulus and argument (a and b are real numbers):
1 i; 2 5i; a + bi(a = 0)
3. Find all values of the following roots and plot t
Fall 2011-MTH5130-Theory of a Complex Variable-Test 2
1. Evaluate the integral (without use of residues)
+
dx
.
+1
x4
0
2. Expand the function
z
z 2 2z + 5
in a Taylor series at the point z = 1 and nd the radius of convergence. What is
f (100) (1)?
f (z )