185hw3-1.pdf - Math 185 Homework 3 \u2013 Due 09\/22 Peter Koroteev 1 Prove that the functions f(z and f(\u00af z are simultaneously holomorphic 2 Show that a

185hw3-1.pdf - Math 185 Homework 3 – Due 09/22 Peter...

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Math 185 Homework 3 – Due 09/22 Peter Koroteev 1. Prove that the functions f ( z ) and f z ) are simultaneously holomorphic. 2. Show that a harmonic function satisfies a formal partial differential equation 2 u ∂z∂ ¯ z = 0 . 3. Show that 1 + 2 z + 3 z 2 + · · · + nz n - 1 = 1 - z n (1 - z ) 2 - nz n 1 - z . 4. Let f ( z ) = az 2 + bz ¯ z + c ¯ z 2 , where a, b and c are fixed complex numbers. Show that f ( z ) is complex differentiable at z iff bz + 2 c ¯ z = 0. Where is f ( z ) holomorphic? 5. Derive the polar form of the Cauchy-Riemann equations for u and v ∂u ∂r = 1 r ∂v ∂θ , ∂u ∂θ = - r ∂v ∂r , for the holomorphic function f ( z ) = u ( r, θ ) + iv ( r, θ ). 6. Show that if f and ¯ f are both holomorphic in a domain
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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

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