Unformatted text preview: Qualifying Exam
MA 530 August 2003
Professor Donnelly I. The ﬁrst four questions are each worth ten points.
√
1. Write (1 − i 3)85 in the form a + ib.
2. Let Γ be the contour consisting of two circles z − 1 = 2, counterclockwise, and
z − 1 = 1, clockwise. Find the value of the winding number in each component of
the complement of Γ. 3. How many zeroes does 2z 2 − ez/2 have in z  < 1?. 4. Find the ﬁrst ﬁve nonzero terms in the Taylor series of sin z
, about z = 0.
z2 + 1 II. The remaining four questions are each worth ﬁfteen points.
5. Give an example of a harmonic function, which is not the real part of a holomorphic
function. Explain why your answer is correct. 6. Suppose f is a holomorphic mapping from the unit disc to itself. Show that, for all
a < 1,
f (a)
1
≤
2
1 − f (a)
1 − a2 7. If u is harmonic and bounded in 0 < z  < ρ, show that the origin is a removable
singularity, in the sense that u becomes harmonic in z  < ρ, when u(0) is properly
deﬁned. 8. By making the substitution x = iy and noticing that y → ∞ as x → ∞, we formally
∞
∞
xdx
ydy
transform the integral
into the integral −
, which is the
4+4
4+4
x
y
0
0
negative of the original integral. Hence, it appears that the value of the integral is
zero. But the true value of the integral is found by elementary methods to be π/8.
What is the fallacy? Explain using complex analysis. ...
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 Spring '09

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