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**Unformatted text preview: **U ⊂ C then f is a constant function. 7. Derive Laplace equation in polar coordinates. Show that function f( z ) = log | z | is harmonic on the punctured plane C \{} .8. Show that the following functions are harmonic and find their harmonic conjugates (a) x 2-y 2 ,(b) sinh x · sin y , (c) ex 2-y 2 cos(2 xy ) . 9. Show that if h ( z) is a complex valued harmonic function (solution of the Laplace equation) such that zh ( z ) is also harmonic, then h (z ) is holomorphic....

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- Fall '07
- Lim
- Derivative, Complex number, Holomorphic function, 2Z, Peter Koroteev