MATH502_HW1 - Serge Ballif MATH 502 Homework 1 January 25,...

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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 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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MATH502_HW1 - Serge Ballif MATH 502 Homework 1 January 25,...

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