**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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