FinalPractice - 12 Show that e x sin y is harmonic on C 13...

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1 Math 375 Final Practice Questions 1. Find all solutions to z 6 = - 9. 2. Show that | z + w | 2 - | z - w | 2 = 4 Re ( z w ) for any z, w C . 3. Show that if f and f are both holomorphic in a region G , then f is constant in G . 4. State the Cauchy-Riemann equations. 5. Find the M¨obius transformations satisfying each of the following. Write your answers in standard form: (a) 0 7→ 0, 1 7→ 1, ∞ 7→ 2 (b) i 7→ - 1, 2 i 7→ - 2, 0 7→ 0. 6. Find the fixed points in C of f ( z ) = z 2 2 z + i . 7. Find all solutions to the equation exp( z ) = πi . 8. Find the length of the curve γ parametrized by γ ( t ) = e it + iπe it for 0 t π . 9. Compute R γ z 2 dz , where γ is the semicircle from 1 through i to - 1. 10. Prove Liouville’s Theorem: Every bounded entire function is constant. 11. State the First Fundamental Theorem of Calculus.
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Unformatted text preview: 12. Show that e x sin y is harmonic on C . 13. Prove that Z is complete. 14. Derive a formula for the product of two power series. 15. State the Maximum-Modulus Theorem. 16. Find the Taylor series about 0 for the following functions. (a) ( z 2-1) e z (b) 1 1+ z (c) 1 e z 17. Find the multiplicities of all zeros of (1 + z 2 ) 3 . 18. Prove that if f is entire and constant on the disk D 1 (0) then f is constant. 19. Let γ be the circle of radius 3 centered at 0. Compute the following integrals. (a) R γ 1 z 2 +1 dz (b) R γ sin z z 2 dz 20. Find the poles of z 1-e z and determine their orders. 1...
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