math185-hw3 - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 3 The real and imaginary parts of z C are denoted by Re( z ) and Im( z ) respectively. If C , we let R := { ( x, y ) R 2 | x + iy } . We will use || to denote the complex modulus in C and kk to denote the vector 2-norm in R 2 . For f : C , we say that z is a zero of f if f ( z ) = 0; we say that f is identically zero on , denoted f 0, if f ( z ) = 0 for all z . You may use without proof any results that have been proved in the lectures. 1. Let f : C C be expressed in the usual form f = u + iv where u, v : R 2 R . Define the function F : R 2 R 2 , F ( x, y ) = ( u ( x, y ) , v ( x, y )) . Show that the following are equivalent. (i) f is differentiable at z = x + iy as a function from C C . (ii) F is differentiable at ( x, y ) as a function from R 2 R 2 and the Jacobian matrix J F ( x, y ) R 2 2 is the composition of a dilation and a rotation, ie....
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math185-hw3 - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

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