Math 656 * Spring 2011 Homework #1 Due date: January 24 REVIEW PROBLEMS: 1. Approximate (using leading two terms in the Taylor series):
cos 0.04 ?
Do not differentiate: simply combine the series for cos x with a series for 1 x 2. Examine the limit of each
Math 656 * Homework 10
Due Monday February 28, 2011
1. Compare
(Im z ) 2 dz and
C1 C2
(Im z )
2
dz , for the contours shown below (C3 is a
C3
semi-circle of radius 1 around x=1)
y
C1
z=0
z=2+i
C2
x
z=2
C3
2. Sketch the smooth arc C defined below, and cho
Math 656 * Homework 10
Due Monday February 28, 2011
1. Compare
(Im z )
(Im z ) 2 dz and
C1 C2
2
dz , for the contours shown below (C3 is a
C3
semi-circle of radius 1 around x=1)
y
z=2+i
C1
C2
z=0
z=2
x
C3
C1 : z (t ) t (2 i ), 0 t 1 dz (2 i )dt , Im z t
Math 656 * Homework 9 Due Thursday February 24, 2011
1. Solve the Laplace's equation in the upper half-plane, with boundary conditions sketched below, as a real or imaginary part of a linear combination of logarithms. Name one branch of the logarithm whic
Math 656 * Homework 9
Due Thursday February 24, 2011
1. Solve the Laplaces equation in the upper half-plane, with boundary conditions
sketched below, as a real or imaginary part of a linear combination of logarithms.
y
=2
x= 0
=0
x=1
x
=1
The solution c
Math 656 * Homework 8 Due Thursday February 21, 2011
1. Find all values of z, and sketch them as points in the complex plane: a) z (2i )3i b) sin z i c) 2e z 2i i 1
2. Derive the following formula, using the definition of tan z in terms of complex exponen
Math 656 * Homework 8
Due Thursday February 21, 2011
1. Find all values of z, and sketch them as points in the complex plane:
a) z (2i )3i
b) sin z i
(a) z (2i ) e3i log(2i ) e
3i
c) 2e z 2i i 1
3i ln 2 i i 2 n
2
e
i (3ln 2)
3
6 n
2
2i 2exp i i 2 n
2
Math 656 * Homework 7
Due Thursday February 17, 2011
1. Problem 1(b) of homework 6 allows us to see that the solution to the Laplaces
equation in a semi-infinite strip sketched below should equal the real or imaginary
part of an analytic function f ( z )
Math 656 * Homework 7 1. Problem 1(b) of homework 6 allows us to see that the solution to the Laplace's equation in a semi-infinite strip sketched below should equal the real or imaginary part of an analytic function f ( z ) A exp(a z ) B exp(b z ) (where
Math 656 * Homework 6 Solution 1. For each of the following two analytic functions f = u + i v, plot the level curves u=0 and v=0, and also make a rough sketch of the level curves u=k and v=k for non-zero constant k
a) f ( z ) z 3 r 3 exp(3i ) r 3 cos(3 )
Math 656 * Homework 5
Due Thursday February 10, 2011
1. Which of the following functions satisfy the Cauchy-Riemann equations? If the
function does satisfy the C-R equations, express it in terms of z alone:
a) f ( x, y ) y 3 3 x 2 y 2 y i ( x3 3 xy 2 2 x)
Math 656 * Homework 4 Due Monday February 7, 2011 1. Prove that the reciprocal rule holds for complex differentiation, at all points where f(z) is analytic and non-zero (use the usual limit definition of derivative): 1 1 f ( z ) f ( z h) 1 d 1 f ( z h) f
Math 656 * Homework 3
1. Show that any punctured ball B '( z0 , r ) is an open set. What are its limit points? What is its
closure?
To prove that B '( z0 , r ) is an open set, we have to prove that all of its points are interior points:
namely, for each p
Math 656 * Spring 2011
Homework #2
Due date: January 27
1. Coinsider any non-zero complex number (0 z )
Prove that | z | | ( z ) | | ( z ) | 2 | z |
Show that either (but not both!) of these inequalities may be an equality
i) | z | | ( z ) | | ( z ) |
Thi
Math 656 * Homework 11 Due Thursday March3, 2011 1. Use appropriate branches to evaluate the following two integrals over any simple contour connecting points 2 and 2 in the upper half-plane:
(a) One can use the principal branch of logarithm (contour does