Ploni Almoni
0.5772156649
MATH 301, Homework 0
Homework 0 does not contain any mathematics it is just for you to practice using
latex. All I want you to do is to try to reproduce this document as well as you can. You do
not have to hand it in. I dont mind
Ploni Almoni
0.5772156649
MATH 301, Homework 10 solutions
1. Weve seen in class that that the Fourier transform of f (x) = 1/(x2 +1) is f (k) = e|k| .
Use properties of the Fourier transform to compute the Fourier transforms of
x
+ 1)2
eiax
d.
(x b)2 + 1
MATH 301, Homework 5
1. Compute p.v.
Do not hand in
x
dx
0 x2 4
for 1 < < 1.
We use a branch of z = r ei with range of angles [0, 2], and integrate along a
contour as shown.
We have = CR + C + A1 + A2 + L1 + L2 , with segments as marked in the contour. N
MATH 301, Homework 7solutions
1. Find the image under the map w = ez of the following domains.
(a) cfw_Rez > 0: The complement ofhte unit disc: cfw_|w| > 1.
(b) cfw_Rez < 0: The unit disc.
(c) cfw_0 < Imz < : The upper half plane.
(d) cfw_0 < Imz < , Rez
MATH 301, Homework 6
Due Feb. 26, 11:00
1. Find a branch of (z 3 + 8)1/3 that is analytic inside the ball cfw_|z| < 2.
1
We simply take the principle brance of w1/3 = e 3 Logw with w = z 3 + 8. If |z| < 2 then
|z 3 | < 8 and so z 3 + 8 is not real negativ
MATH 301, Homework 8solutions
7.3 #2 The line Imz = 0 passes through a pole, so it is mapped to a line. The image contains
f (1) = 1 i and f (1) = 1 + i, so the image is the line cfw_Rez = 1.
The line cfw_Imz = 1 is mapped to a circle. It passes through f
MATH 301, Homework 1 solutions
Omer Angel
From Section 5.6
1a. The only singularities are at 0, 1 At 0 there is a pole of order 2 and at Your solution
to 1a here. At 1 there is a simple pole.
1b. There is an essential singularity at z = 0.
1c. There are s
solutions
MATH 301, Homework 11 solutions
1. Find the solution to the heat equation ut (x, t) = uxx (x, t) for t 0 with
2
2
u(x, 0) = ex /(2 ) .
Using the convolution with the fundamental solution
1
u(x, t) =
2 t
ey
2 /(2 2 )
2 /4t
e(xy)
2
1
ex /4t
2 t
MATH 301, Homework 9 solutions
1. Solve the Laplace equation in the half circle cfw_|z| < 2, Im(z) > 0 with boundary values 0 on the diameter and 1 on the circle.
z+2
We note that w = z2 maps the domain to the quarter plane cfw_x, y > 0, with the
diameter
Math 301 Final Examination April 24, 2010
THIS EXAM HAS 8 QUESTIONS.
YOU ARE PERMITTED ONE SHEET OF NOTES (DOUBLE SIDED).
NO CELLPHONES, BOOKS OR CALCULATORS.
1. (10pts) Use contour integration to calculate
0
of every step.
sin x
dx, including a brief exp
The University of British Columbia
Final Examination - April, 2011
Mathematics 301
Closed book examination
Last Name:
Time: 2.5 hours
, First:
Signature
Student Number
Special Instructions:
No books, notes or calculators are allowed.
Include explanations
1
University of British Columbia
Math 301, Section 201
Final exam
Date: April 18, 2012
Time: 8:30 - 11:00am
Name (print):
Student ID Number:
Signature:
Instructor: Richard Froese
Instructions:
1. No notes, books or calculators are allowed. A summary sheet
1
University of British Columbia
Math 301, Section 201
Final exam
Date: April 20, 2013
Time: 8:30 - 11:00am
Name (print):
Student ID Number:
Signature:
Instructor: Richard Froese
Instructions:
1. No notes, books or calculators are allowed. A summary sheet