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homework108c1

# homework108c1 - that d dz g z = f z Hint The imaginary part...

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HOMEWORK 1 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, APRIL 7, 2009 (1) Let f be holomorphic and non-vanishing. Show that log( | f | ) is harmonic. (Here we use the usual deﬁnition of log : R > 0 R ). (2) Show that in polar coordinates the Cauchy-Riemann equations have the form ∂u ∂r = 1 r ∂v ∂θ , 1 r ∂u ∂θ = - ∂v ∂r , where as before f = u + iv is a holomorphic function. Now show that (for z = re ) z = re iθ/ 2 is holomorphic in the region r > 0 and - π < θ < π . Give a simple sketch of the complement of this region. (3) Show that f ( z ) = 1 /z is holomorphic on the annulus A (1 , 2) = { z C | 1 < | z | < 2 } . Show it has no holomorphic anti-derivative (i.e. a function g such
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Unformatted text preview: that d dz g ( z ) = f ( z )). Hint: The imaginary part of g can only be deﬁned up to a constant. (4) Calculate the following integrals using the deﬁnition of contour integral: (a) H γ z n dz for n ∈ Z (also negative numbers!) and γ = C (0 , 1), the circle with center 0 and radius 1, traversed in a positive direction. (b) H γ z n dz for n ∈ Z and γ = C (2 , 1), again traversed in a positive direc-tion. (c) For | a | < r < | b | show that H γ 1 ( z-a )( z-b ) dz = 2 πi a-b for γ = C (0 ,r ), traversed in positive direction. 1...
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