Math 300 Assignment 8: due Mar 18th in class
1. Find the Laurent series for sin z/z 3 for |z| > 0.
2. Find the Laurent series for
z+1
z(z−4)2
in 0 < |z − 4| < 4.
3. Find and classify the isolated singularities of each of the following functions: (a) 1/(ez − 1);
1
sin z
(b) cos(1/z)
sin(1/z) ; (c) z 2 − z .
4. Define the function h(z) =
1
sin z
−
1
z
+
2z
.
z 2 −π 2
a) Show h(z) is analytic in the disk |z| < 2π except for the removable singularities at z = 0,
z = ±π.
b) Find the first four terms of the Taylor series about z = 0 for h(z). What is the radius of
convergence for this series?
c) Use the result of (b) to obtain the first few coefficients (for positive and negative indices
|j| ≤ 4) in the Laurent series expansion for sin1 z , valid in the annulus π < |z| < 2π.
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The following questions are for fun only, they will not be graded. Try to solve these problems!
5. Let m ≥ 1 be an integer. Show that if f is entire and |f (z)| ≥ |z|m for |z| ≥ 1. Then f must
be a polynomial of degree at least m.
6. Suppose f and g are both analytic in D = {z :
maximum on the boundary ∂D = {z : |z| = 1}.
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|z| ≤ 1}. Show that |f (z)| + |g(z)| take its