MATH 3C03: Home Assignment # 2
Due to: October 9, 2014
Problem 1: Find the unique solution of the secondorder dierential equation
d2 y
dy
5 + 6y = 0,
2
dt
dt
satisfying the initial values y(0) = 1 and y (0) = 0.
Problem 2: Find the most general solution
Math 3C03
M. MinOo
Short Answers to Assignment #3
1. Expand f (x) = x(1 x); 1 x +1 in terms of Legendre polynomials and verify Parsevals
identity.
1
x(1 x) = P1 (x) x2 = 3 P0 (x) + P1 (x) 2 P2 (x)
3
+1 2
1 x (1
x)2 dx =
16
15
which is equal to
1
3 P0
3
Math 3C03
M. MinOo
Short Answers to Assignment #5
1. Show that the function (x, y, z) = x2 + y 2 2z 2 represents a simultaneous eigenstate of L2
and Lz . What are the eigenvalues? Find a wave function with the same eigenvalue for L2 with the
maximum poss
Math 3C03
Short Answers to Assignment #2
#1.
f (t) = cosh(t 1) for 0 t 1, extended periodically as an even function with period 2.
+1
cosh(t 1) cos(nt) dt =
an =
1
2 sinh(1)
2 n2 + 1
so
cosh(t 1) = sinh(1) + 2
n=0
sinh(1)
cos(nt)
2 n2 + 1
is the require
Math 3C03
M. MinOo
Short Answers to Assignment #4
1. Compute:
+
2
x2 (Hn (x)2 ex dx
where Hn (x) is the
nth
Hermite polynomial.
The integral is equal to xHn (x)
We use the following facts:
2
with respect to the inner product < f g >=
+
x2 dx.
f (x)g(x)
Math 3C03
M. MinOo
Assignment #3
Due: Friday, October 26th, 2012 in class at the beginning of the lecture
1. Expand f (x) = x(1x); 1 x +1 in terms of Legendre polynomials and verify Parsevals
identity.
2.
Do problem 16.14 on page 552 in the textbook.
3.
Math 3C03
M. MinOo
Assignment #2
Due: Friday, October 5th, 2012 in class at the beginning of the lecture
1.
Do problem 12.16 on page 429 in the textbook.
2. Compute the Fourier sine series of the odd function f (x) = x3 4x; 2 x 2 (periodically
extended w
Math 3C03
M. MinOo
Assignment #4
Due: Friday, November 9th, 2012 in class at the beginning of the lecture
1.
Compute:
+
2
x2 (Hn (x)2 ex dx
where Hn (x) is the nth Hermite polynomial.
2. Find the electric potential inside and outside a spherical capacito
Math 3C03
M. MinOo
Assignment #5
Due: Friday, November 23rd, 2012 in class at the beginning of the lecture
1. Show that the function (x, y, z) = x2 + y 2 2z 2 represents a simultaneous eigenstate of L2
and Lz . What are the eigenvalues? Find a wave funct
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Math 3C03
M. MinOo
Assignment #1
Due: Friday, September 21st, 2012 in class at the beginning of the lecture
1.
Do problem 8.38 on page 313 in the textbook.
2.
Do problem 9.1 on page 329 in the textbook.
3.
Show that
x
a
b
2
2
x
a
b2 = (b a)(x a)(x b)(x +
MATH 3C03: Home Assignment # 3
Due to: October 28, 2014
Problem 1: Consider the Hermite dierential equation
d2 y
dy
2x + 2ny = 0,
2
dx
dx
where n is an integer. Look for two linearly independent solutions in the power series
form and identify which solut
Math 3C03
M. MinOo
Assignment #2
Due: Thursday, October 3rd, 2013 in class at the beginning of the lecture
1.
Do problem 12.14 on page 429 in the textbook.
2. Evaluate the Fourier transform (k) for k = (0, 0, k) (in the direction of the axis of symmetry)
Math 3C03
M. MinOo
Assignment #5
Due: Thursday, November 21st, 2013 in class at the beginning of the lecture
1.
Solve the heat equation
1 2
u(x, t) = 2 2 u(x, t)
t
x
on the real line R with initial condition:
u(x, 0) =
2.
1 for x 1
0 otherwise
(i) Fin
MATH 3C03: Home Assignment # 2
Due to: October 9, 2014
Problem 1: Find the unique solution of the secondorder dierential equation
d2 y
dy
5 + 6y = 0,
2
dt
dt
satisfying the initial values y(0) = 1 and y (0) = 0.
Problem 2: Find the most general solution
MATH 3C03: Home Assignment # 4
Due to: November 13, 2014
Problem 1: Express the function f (x) = cos(x) given for x (0, 1) as Fourier sine and
cosine series. Plot the Fourier sine and cosine series for x (1, 1).
Problem 2: Use the Fourier series to solve
MATH 3C03: Home Assignment # 5
Due to: November 27, 2014
Problem 1: Solve the Laplace equation u = 0 inside a sphere of radius R under the
boundary condition ur=R = 5 cos2 (), where r is the radial variable and is the latitudinal
angle in spherical coord
MATH 3C03
Short Answers to Test # 1
# 1. State whether the following statements are TRUE or FALSE (no explanations needed)
(i) Every square matrix is diagonalisable.
FALSE
(iii) For any matrix A, all the nonzero eigenvalues of AAT and AT A are the same.
Math 3C03
M. MinOo
Assignment #3
Due: Thursday, October 24th, 2013 in class at the beginning of the lecture
1.
Do problem 16.14 on page 552 in the textbook.
2.
Expand f (x) = x2 (1 x)2 in terms of Legendre polynomials and verify Parsevals identity.
3.
Do
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MATH 3C03 (Term1, 2012/13)
Instructor: M. MinOo
Marker: A. Moghrabi
Assignment #:
Due date:
Last Name :
First Name :
Student ID # :
Q
#1
#2
#3
#4
#5 (bonus)
Sum
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1
Fourier Transform
The Fourier transform F(f ) = f of a function f : Rn C is defined by:
n
F(f )(k) = f (k) = (2) 2
f (x)eikx dx
k Rn
where kx =
Rn
ki xi .
The integral converges if f (x) decays suciently rapidly at infinity. In fact, the Fourier transfo
Math 3C03
M. MinOo
Assignment #2
Due: Thursday, October 3rd, 2013 in class at the beginning of the lecture
1.
Do problem 12.14 on page 429 in the textbook.
b ~k) for ~k = (0, 0, k) (in the direction of the axis of symmetry)
2. Evaluate the Fourier transf
Fourier Series
A Fourier series is a trigonometric series of the form:
+
X
ck eikx =
a0 X
+
(ak cos kx + bk sin kx)
2
k=1
where ck = 12 (ak ibk ) and ck = 21 (ak + ibk ). If the series converges, it will define a realvalued
function which is periodic wit