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Unformatted text preview: 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 CauchyRiemann 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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This note was uploaded on 04/01/2008 for the course MATH 502 taught by Professor Johnridener during the Summer '08 term at Penn State.
 Summer '08
 JOHNRIDENER
 Math

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