math185-hw3sol - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 3 SOLUTIONS 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 k k 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 we have already 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. J F ( x,y ) = r cos sin - sin cos for some r 0 and- / 2 / 2. How are r and related to u and v ? Solution. By Theorem 2.8 in the lectures, if f satisfies (i), then the Cauchy-Riemann equa- tions u x = v y , u y =- v x must hold at z . In which case, J F ( x,y ) = u x u y v x v y = u x u y- u y u x = r cos sin - sin cos where r = q u 2 x + u 2 y and = arctan( u y /u x ). If F satisfies (ii), then we let u x = v y = r cos and u y =- v x = r sin . In which case, f ( z ) = u x + iv x = r (cos - i sin ) . In either case, we have the equality f ( z + h )- f ( z ) h- f ( z ) = | f ( z + h )- f ( z )- f ( z ) h | | h | = k F ( x + h 1 ,y + h 2 )- F ( x,y )- J F ( x,y ) h k k h k where h = h 1 + ih 2 C and h = ( h 1 ,h 2 ) > R 2 . The denominators are equal because | h | = p h 2 1 + h 2 2 = k h k . The numerators are equal because f ( z ) h = r (cos - i sin )( h 1 + ih 2 ) = r ( h 1 cos + h 2 sin ) + ir ( h 2 cos - h 1 sin ) and J F ( x,y ) h = r cos sin - sin cos h 1 h 2 = r ( h 1 cos + h 2 sin ) r ( h 2 cos - h 1 sin ) . In the other words, the limit as h 0 in C exists iff the limit as h in R 2 exists....
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math185-hw3sol - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

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