Math 303 - Complex Variables
Homework due March 30
Recall that Cauchy’s Integral Formula gave a simple way to compute certain loop integrals. Let Γ
be a positively oriented simple closed contour. If fis analytic in some simply connected domain D
containing Γ and z0is any point interior to Γ, then
2πif (z0) = ZΓ
f(z)
z−z0
dz.
As a consequence, we also have the more general formula
2πi
(n−1)!f(n−1)(z0) = ZΓ
f(z)
(z−z0)ndz.
Question 1. Let Cbe the circle |z|= 2 traversed once counterclockwise. Use the Cauchy Integral
Formula to compute the following loop integrals.
(a) ZC
sin 3z
z−π
2
dz
(b) ZC
zez
2z−3dz
(c) ZC
cos z
z3+ 9zdz
(d) ZC
5z2+ 2z+ 1
(z−i)3dz
(e) ZC
e−z
(z+ 1)2dz
(f) ZC
sin z
z2(z−4) dz
Question 2. Let fbe analytic inside and on the simple closed contour Γ. What is the value of
ZΓ
f(z)
z−z0
dz
when z0lies outside Γ?
Question 3. Let fand gbe analytic inside and on the simple loop Γ. Prove that if f(z) = g(z) for
all zon Γ, then f(z) = g(z) for all zinside Γ.
Question 4. Compute
ZC
z+i
z3+ 2z2dz
where Cis
(a) the circle |z|= 1 traversed counterclockwise.
(b) the circle |z+ 2 −i|= 2 traversed once counterclockwise.
(c) the circle |z−2i|= 1 traversed counterclockwise.
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