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homework20111028 - a 7 z 1 − 4 z 3 b e 3 z 2 c z cos z 2...

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Math 417, Fall 2011 Exercise Set #6 Turn in: October 28 1. Evaluate C 2 z + 1 z 4 2 z 2 + 1 dz , where C is the circle | z | = 10 with the positive orientation. 2. Show that if n =1 z n = S , then n =1 z n = S . 3. What is the radius of convergence of the Taylor Series for f ( z ) = 1 z 2 3 z + 2 about z = 0? About z = 3 i ? 4. Find the radius of convergence for the following power series: a) n =1 n z n b) n =1 z n n c) n =1 z n n ! d) n =1 z n 2 5. Find the Taylor series expansion of the following functions around z = 0. State the radius of convergence of the series.
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Unformatted text preview: a) 7 z 1 − 4 z 3 b) e 3 z 2 c) z cos( z 2 ) d) z cosh( z 2 ) 1 2 . 6. Find elementary expressions for the following power series a) ∞ ° n =1 z 3 n +1 b) ∞ ° n =1 n ( n − 1) z n c) ∞ ° n =1 z 2 n n ! 7. Let z = re iθ where 0 < r < 1. Show that a) ∞ ° n =1 r n cos( nθ ) = r cos( θ ) − r 2 1 − 2 r cos( θ ) + r 2 b) ∞ ° n =1 r n sin( nθ ) = r sin( θ ) 1 − 2 r cos( θ ) + r 2...
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