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1. (10pts) Show that u(z,y) = e-3r cos 3y is harmonic and find a harmonic conjugate. 2. (10pts) Find a biholomorphic map between the unit disk A and the region = 3. (20pts) Let a € C be a constant. (a) Find the inverse Laplace transform of F(8)= 3+28 using the Bromwich integral. (b) Find the inverse Laplace transform of 1 s+a using the Bromwich integral. (c) Find the inverse Laplace transform of 1 F(s) (s+a)² using the Bromwich integral. (d) Decompose the rational function 1 s+2s²+s into principal parts and use this together with (b) and (c) to find the inverse Laplace transform. 4. (20pts) Show that do dx = where O and 1 if>>0 sgn = -1 if < < 0. (Hint: First do the case a > 0 and 8>0.) 5. (20pts) Compute .° where a € (-1,1). 1 6. (20pts) Show that the equation Z4 + z+1 = 0 has one solution in each quadrant. Prove that all solutions lie inside the open disk [zec1121<21 2

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