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# prob03 - (a u x,y = xy-x y(b u x,y = e y sin x 6 Find the...

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Problem Set 3 Due February 12 th , 2008 1. Let g ( x + yi ) = 3 x 2 + 2 x - 3 y 2 - 1 + i (6 xy + 2 y ). Show that g is an analytic function on C . 2. Show that if f is analytic in a domain D and either Re f ( z ) of Im f ( z ) is a constant in D , then f ( z ) must be a constant. 3. Suppose that both f ( z ) and f ( z ) are analytic in a domain D , then f ( z ) is constant. 4. Find a function ϕ ( x, y ) harmonic in the upper half-plane Im z > 0 and continuous on Im z 0 such that (a) ϕ ( x, 0) = x 2 + 5 x + 1 for all x . (b) ϕ ( x, 0) = x 3 / ( x 2 + 9) for all x . 5. Find the harmonic conjugate of following function.
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Unformatted text preview: (a) u ( x,y ) = xy-x + y . (b) u ( x,y ) = e y sin x . 6. Find the partial fraction for following rational functions: (a) 3 + i z ( z + 1)( z + 2) . (b) 3 z + i z 3 + z (c) z ( z 2 + z + 1) 2 (d) 5 z 4 + 3 z 2 + 1 2 z 2 + 3 z + 1 7. Let P ( z ) be a polynomial. Prove that all the coeﬃcient of P ( z ) are real if and only if P ( x ) is real for any real number x ....
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