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MATH502_HW1 - Serge Ballif MATH 502 Homework 1(1 Let f = u...

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Serge Ballif MATH 502 Homework 1 January 25, 2008 (1) Let f = u + iv be a holomorphic function of z = x + iy . Show that the function g = log | f | 2 = log( u 2 + v 2 ) satisfies Laplaces equation 2 g ∂x 2 + 2 g ∂y 2 = 0 in a neighborhood of any point where f is a nonzero. (You may assume that u and v are twice continuously differen- tiable. (Try to do this in two different ways: once by a brute force calculation using the Cauchy-Riemann equations and once by using the properties of the locally defined holomorphic logarithm function, discussed in lecture 5) Using the “brute force” method, we calculate a first partial derivative of g with respect to x ∂g ∂x = ∂x [log( u 2 + v 2 )] = ∂x [ u 2 + v 2 ] u 2 + v 2 = 2 u ∂u ∂x + 2 v ∂v ∂x u 2 + v 2 . Then the second partial with respect to x is 2 g ∂x 2 = ∂x " 2 u ∂u ∂x + 2 v ∂v ∂x u 2 + v 2 # = 2 ( ∂u ∂x ) 2 + 2 u 2 u ∂x 2 + 2 ( ∂v ∂x ) 2 + 2 v 2 v ∂x 2 ( u 2 + v 2 ) - ( 2 u ∂u ∂x + 2 v ∂v ∂x ) 2 ( u 2 + v 2 ) 2 = 2 u 3 2 u ∂x 2 + 2 u 2 ( ∂v ∂x ) 2 + 2 u 2 v 2 v ∂x 2 + 2 v 2 ( ∂u ∂x ) 2 + 2 uv 2 2 u ∂x 2 + 2 v 3 2 v ∂x 2 - 4 uv ∂u ∂x ∂v ∂x ( u 2 + v 2 ) 2 . By symmetry 2 g ∂y 2 = 2 u 3 2 u ∂y 2 + 2 u 2 ∂v ∂y 2 + 2 u 2 v 2 v ∂y 2 + 2 v 2 ∂u ∂y 2 + 2 uv 2 2 u ∂y 2 + 2 v 3 2 v ∂y 2 - 4 uv ∂u ∂y ∂v ∂y ( u 2 + v 2 ) 2 .
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