Math 310, Fall 2012
Homework 10 Solutions
Due: November 16
Be sure to read the homework guidelines before completing this assignment.
(1) Suppose that f is analytic in a domain G, that z0 G and that f (z0 ) = f (z0 ) = = f (m) (z0 ) = 0.
Prove that the fu
Problem Set 4: Mathematics 310
Instructor: Stanley Chang
1. Let F : R2 R2 be a vector field. We say that the field is incompressible if the divergence
F is zero, and we say that the field is irrotational if the curl F is zero. We say that
F is an ideal f
Problem Set 3: Mathematics 310
Read pages 196-204 of the Rudin text.
1. Let f : Rn Rm be a function, and let its i-th coordinate be given by fi : Rn R. Prove that
f is continuous at a = (a1 , . . . , an ) iff all the fi are continuous at a.
2. Recall that
Problem Set 6: Mathematics 310
Instructor: Stanley Chang
1. Find all removable singularities, zeroes and poles of the given functions. Find the order of the
zeroes and poles. Write down the analytic extension of f in the case of a removable singularity.
(
Problem Set 2: Mathematics 310
1. For the following subsets of C, state whether it is (i) open, closed or neither, (ii) connected.
No explanation is required. Then identify (iii) its boundary and (iv) its interior in set notation
and xy-coordinates.
(a) A
EASTRIDGE GYM SCHEDULE - DECEMBER
Revised: 12.15.16
Thursday
Gym Closures:
December 16 & 17
Breakfast with Santa
December 24 (close at 5pm)
Christmas Eve
December 25
Christmas Day
December 31 (close at 5pm)
New Year's Eve
Thursday
Open Gym/Basketball: Res
Math 310, Fall 2012
Homework 1 Solutions
Due: September 7
Be sure to read the homework guidelines before completing this assignment.
(1) For z, w C, show that
|z + w|2 = |z|2 + 2Re(z w) + |w|2
and
|z w|2 = |z|2 2Re(z w) + |w|2 .
Solution. For the rst expr
Math 310, Fall 2012
Homework 3 Solutions
Due: September 21
Be sure to read the homework guidelines before completing this assignment.
Read sections 16-22 of the text.
1
(1) Use the denition of the derivative to prove that for the function f : C \ cfw_0 C
Math 310, Fall 2012
Homework 5 Solutions
Due: October 5
Be sure to read the homework guidelines before completing this assignment.
Read sections 30-35 of the text.
(1) Suppose that T is a Mobius transformation that xes exactly the points z1 , z2 C. If S
Math 310, Fall 2012
Homework 2
Due: September 14
Be sure to read the homework guidelines before completing this assignment.
(1) Suppose n > 1. Let n = cos
2
n
+ i sin
2
n
.
0
1
2
n1
(a) Prove n + n + n + + n = 0.
Solution. Notice that by deMoivres theorem
Math 310, Fall 2012
Homework 6
Due: October 19
Be sure to read the homework guidelines before completing this assignment.
Read sections 35-40 of the text.
(1) Suppose that C is a simple closed contour oriented in the counterclockwise direction. Prove tha
Math 310, Fall 2012
Homework 4
Due: September 28
Be sure to read the homework guidelines before completing this assignment.
Read Chapter 3 of the text.
(1) Prove the complex analogs of the usual formulae for sin( + ) and cos( + ).
Solution. We know that
Math 310, Fall 2012
Homework 7
Due: October 26
Be sure to read the homework guidelines before completing this assignment.
Read sections 41-43 of the text.
(1) Calculate the following integrals:
(a)
sin(cos(z)
dz, where (t) = eit for t [0, 2];
z3
Solution
Math 310, Fall 2012
Homework 12
Due: December 11
Be sure to read the homework guidelines before completing this assignment. For all improper integrals, you
are free to assume that these integrals do converge.
(1) Evaluate
x2
dx where a and b are distinct
Math 310, Fall 2012
Homework 11
Due: November 30
Be sure to read the homework guidelines before completing this assignment.
Read sections 56-59 in the text.
z
.
1)(ez 2)
Solution. Since the numerator and denominator are both entire functions, the quotie
Math 310, Fall 2012
Homework 9 Solutions
Due: November 9
Be sure to read the homework guidelines before completing this assignment.
Read sections 48-51 of the text.
(1) Give the MacLaurin series for f (z) =
1
. For what values of z is this representation
Math 310, Fall 2012
Homework 8 Solutions
Due: November 2
Be sure to read the homework guidelines before completing this assignment.
Read sections 43-45 of the text.
(1) Suppose that is a simple closed curve, and that f and g are functions which are analy
Problem Set 7: Mathematics 310
Instructor: Stanley Chang
1. Use a semicircular contour to compute the following.
Z
x2
(a)
dx
4
1 + x
Z
x2
(b)
dx
4
2
x + 4x + 3
2. Compute the following by changing the trigonometric function to an exponential.
Z
cos x