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MA530_JAN97

# MA530_JAN97 - C of the following functions a 1 e z-1-1 z b...

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MA 530 Qualifying Examination January 1997 Name: 1. Let r be the radius of convergence of a n z n ,and ρ be the radius of convergence of b n z n .S e t c n = a 0 b n + a 1 b n - 1 + ··· + a n b 0 for n =0 , 1 ,.... What can the radius of convergence R of the series c n z n be? Describe all possibilities. 2. For what R does there exist a one-to-one analytic map f R from the left half-plane { z C : < ez< 0 } onto the disk D R = { z C : | z | <R } such that f R ( - 3) = 0 and f 0 R ( - 3) = 1 + i ?F ind f R for those R for which it exists and prove that for all other R ’s it does not exist. 3. How many zeros does the polynomial z 10 - z
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Unformatted text preview: C of the following functions: a) 1 e z-1-1 z b) e z-1 z c) z ( e 1 z-1) d) e tan 1 z e) z 2 sin z z +1 . 5. Evaluate the integral Z ∞ x 4 x 6 + 1 dx. 6. Find all possible values of Z 1 dz 1 + z 2 for all paths going from 0 to 1 such that the integral converges. 7. Let P be a harmonic polynomial of two real variables. Show that the conjugate har-monic function is also a polynomial. 8. Find the inﬁnite product expansion for e z-1....
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