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Unformatted text preview: MATH 530 Qualifying Exam
January 2011, G. Buzzard, S. Bell
1. (a) (10 points) Deﬁne
π f (z ) :=
−π cos2 t
dt.
z − sin t Use the diﬀerence quotient deﬁnition of complex derivative to show that f is holomorphic
on C − [−1, 1] and to write f (z ) as an integral.
(b) (10 points) Can f (z ) be extended to [−1, 1] to make f (z ) an entire function? If so,
describe how to extend f . If not, prove that no such extension exists. 2. (20 points) Let
n Pn (z ) =
k=0 zk
.
k! Prove that for a given R > 0, there exists a positive integer N such that if n ≥ N , then
Pn (z ) has exactly n zeroes in {z : z  > R}. 3. (20 points) Suppose that f has a pole at z0 and that g is holomorphic on C − D (so
that g has an isolated singularity at ∞). Give necessary and suﬃcient conditions on the
singularity of g at ∞ so that g (f (z )) has a pole at z0 and prove that your conditions are
necessary and suﬃcient. 4. (a) (5 points) Suppose f has a pole of order m at z0 . Use the Laurent series expansion
of f around z0 to derive the formula for the residue of f at z0 in terms of derivatives of
(z − z0 )m f (z ).
(b) (15 points) Compute ∞
0 ln x
dx.
+ 1)2 (x2 5. (20 points) Prove or disprove that there is a sequence of polynomials {pn (z )}∞ so
n=1
that pn (z ) converges uniformly on the unit circle, {z : z  = 1}, to the function f (z ) = (z )2 . ...
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This document was uploaded on 01/25/2012.
 Spring '09
 Derivative

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