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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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