Math 656 * Homework 18
Due Monday April 11, 2011
1. Calculate the following integrals over indicated circular contours:
a)
b)
c)
sin z
dz
| z| =1 z sinh z
2
dz
2
| z | = 4 ( z + 1) sin z
log p ( z 4)
| z| = 3
2. Calculate
z2 + 2z + i
| z| =3
dz
exp(1/ z )
Math 656 * Homework 18
Due Monday April 11, 2011
1. Calculate the following integrals over indicated circular contours:
The integrand has poles at multiples of i, but only one of those is located at within the
integration contour, the singularity at z=0,
Math 656 * Homework 16
Due Monday April 4, 2011
f ( n ) ( z0 )
(where n0) is
n!
generally not true for a Laurent series expansion of function f ( z ) unless it is a
Taylor series with zero principal part (otherwise it is generally false even if f (z) is
a
Math 656 * Homework 16
Due Monday April 4, 2011
f ( n ) ( z0 )
(where n0) is generally
n!
not true for a Laurent series expansion of function f ( z ) unless it is a Taylor series with zero
principal part (otherwise it is generally false even if f (z) is a
Math 656 * Homework 14
Due Monday March 28, 2011
1
1. Where does the sequence
converge? Where does it converge
1 + n z n =0
uniformly?
2. Find the radii of convergence of the following series using appropriate tests:
n2 en
zn
n =1
(a)
nn n
n! z
n =1
(
Math 656 * Homework 14
Due Monday March 28, 2011
1
1. Where does the sequence
converge? Where does it converge uniformly?
1 + n z n =0
1
1, z = 0
1
1
The sequence
=
converges for z : lim
n x 1 + n z 0, z cfw_0, 1/ n, n +
1 + n z n =0
Note: one sequen
Math 656 * Homework 13
Due Thursday March 24, 2011
1. Show that if a real function of complex variable h(z) is harmonic on and inside a circle of
radius r, then its value at the center of the circle equals its average over the circle
circumference:
2
1
i
Math 656 * Homework 15
Due Monday March 31, 2011
1. Solution to problem 1 will be posted with homework solution #16
The answer is, only if the principal part is zero, in which case the series is a Taylor series
and converges all the way to the center of e
Math 656 * Homework 15
Due Monday March 31, 2011
1. In class we have proven that a function analytic in a ring domain r<|zzo|<R has a
Laurent series representation converging in this domain (and converging uniformly in
domain r < r* |zzo| R* < R):
f ( z )
Math 656 * Homework 12
Due Monday March 7, 2011
1. Give a parametric description of a simple closed contour that follows the
ellipse x2/a2 + y2/b2 = 1, traced anticlockwise (positively oriented). Consider
an integral of funct
Math 656 * Homework 12
Due Monday March 7, 2011
1. Give a parametric description of a simple closed contour that follows the ellipse
x2/a2+y2/b2 = 1, traced anticlockwise (positively oriented). Consider an integral of
function 1/z around the ellipse, and