math185-hw3 - 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

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

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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 Ω C , we let Ω R := { ( x, y ) R 2 | x + iy Ω } . We will use | · | to denote the complex modulus in C and · to denote the vector 2-norm in R 2 . For f : Ω C , we say that z Ω is a zero of f if f ( z ) = 0; we say that f is identically zero on Ω, denoted f 0, if f ( z ) = 0 for all z Ω. You may use without proof any results that have been proved in the lectures. 1.Letf:CCbe expressed in the usual formf=u+ivwhereu, v:R2R.Define thefunction (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 θ sin θ - sin θ cos θ for some r 0 and - π/ 2 θ π/ 2.

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• Fall '07
• Lim
• Math, Calculus, power series representation

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