CalcIVHW8 - f z(Hint Use your answer from problem#2 6 Using...

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Homework #8 1. (Appendix G: #39) Find the cube roots of i and sketch them in the complex plane. 2. Find functions u ( x, y ) and v ( x, y ) from R 2 to R such that z + ( z ) 2 = u ( x, y ) + iv ( x, y ) . 3. Find all possible values of log ± e 2 + e 2 i ² . 4. Compute the derivative of z 2 directly from the definition. (Hint: Don’t break it into real and imaginary parts; instead, make the change of variables h = z - c used in lecture and then simplify the difference quotient f ( c + h ) - f ( c ) h directly by (complex) algebra.) 5. Using the Cauchy-Riemann equations, determine whether or not f ( z ) = z +( z ) 2 is holomorphic. If it is, find
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Unformatted text preview: f ( z ). (Hint: Use your answer from problem #2.) 6. Using the Cauchy-Riemann equations, show that f ( z ) = 1 /z is holomorphic on its domain and that its derivative is-1 /z 2 . (Hint: The real and imaginary parts of 1 /z were given in lecture.) 7. Show that d dz sin z = cos z . (Hint: Write sin z in terms of the exponential function and then use the rules for differentiation and the fact that we know how to differentiate exponential functions.) 8. Let f ( z ) = ze πz . Compute f ( i ). 1...
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This note was uploaded on 07/25/2008 for the course MATH V1202 taught by Professor Neel during the Spring '07 term at Columbia.

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