1. (a) Solve the equation z8 + z4 + 1 = 0
Let f(z) =(z-e^(ipi/4))/(z^4+1). Determine the points of discontinuity of f(z) and rede…ne f(z) to make it
continuous at z = ei pi/4.
(c) Use the de…nition of di¤erentiability to show that f(z) = =z is nowhere di¤erentiable.
d)(z+1)/(z^(5/6)-1+i) . Determine the values of z for which f(z) is holomorphic.
2. Let f(z) = iz2, de…ned on a domain D, such that D = z = x + iy : 0 < x < 1; 0 < y < 1
(a) Express the given function in the form f(z) = u + iv.
(b) Sketch the domain D, determine the boundary of the domain D and Find the image of the each curve of the boundary under the mapping f(z).
3. (a) Suppose
u(x; y) = x3 - kxy2 + 12xy- 12x
for some constant k. Find all values of k for which u is a real part of a holomorphic function.
(b) Show that the following function is nowhere analytic but is di¤erentiable along a curve that you should determine :
f(z) = x2 - x + y + i(y2 - 5y- x)
(c) Let f(z) = ez. Is f(z) a holomorphic function?
4. Show that u(x; y) = ln(x2 + y2) is a harmonic function.
(a) Find the harmonic conjugate v(x; y) of u(x; y):
(b) Find f0(z) where f(z) = u + iv; and u(x; y); v(x; y) are the functions from part a) and b).
5. (a) Find the values of (1- i)2- i:
(b) Find all values of z for which e4z = 3 + 3i:
(c) Find all values of z for which sin z = i:
(d) Prove that cosh2 z + sinh2 z = cosh 2z:
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