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,
1 − |f (a)|
1 − |a|2 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
transform the integral
into the integral −
, which is the
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. ...
View Full Document
This document was uploaded on 01/25/2012.
- Spring '09