M417
Fall 1996
xfs.tex
1.
Find the poles and zeroes, with multiplicities and residues of
f(z) = tan(πz).
2.
For a real, find
lim
y → ±∞
tan(π(a + i y))
3.
For 0 < a < \frac12, what is the change of arg(tan(πz)) along the curve [line]
C
a
= {z(t) = a + it | −∞ ≤ t ≤ +∞}?
Hint: What is the sign of the real part of tan(πz) along C
a
?
4.
For \frac12 < a < 1, what is the change of arg(tan(πz)) along the curve [line]
C
a
= {z(t) = a + it | −∞ ≤ t ≤ +∞}?
Hint: How many zeroes and poles of f(z) are there between C
\tfrac14
and C
\tfrac34
?
5.
For various values of z
0
, find the radius of convergence of the Taylor series for f(z) about z = z
0
.
6.
Discuss the [multiple valued] functions √z and z
√2
.
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On 06 Feb 2017, 13:33.