Fourier Analysis and Boundary Value Problems For ISP
MATH 381

Fall 2013
MATH 381
FOURIER ANALYSIS AND BOUNDARY VALUE PROBLEMS FOR ISP
FINAL FALL 2011
Please write your name on each bluebook, and the number of bluebooks used, if you
use more than one. Reading time starts at 12:00pm and runs until 12:10pm. During this
period
Fourier Analysis and Boundary Value Problems For ISP
MATH 381

Fall 2013
1
Pinsky 1.4.10 (d)
Note that
1
1
=
(a2 n2 )2
2
n=1
1
1
4
(a2 n2 )2
a
n=
1
so to compute n=1 n4 it suces to compute the limit of the right hand side above as a 0. We do this using
the formula from the previous part of the problem:
1
2
1
1
4
(a2 n2 )2
a
n=
Fourier Analysis and Boundary Value Problems For ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 10
Not handed in
1. (Pinsky 5.1.1516)
a. Let a > 0. Show that the Fourier transform of f ( ax ) is f(/ a)/ a.
b. Show that the Fourier transform of f ( x a) is eia f().
a. If g( x ) = f ( ax ), then we make the change of va
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 1
30 September 2013
1. (Pinsky 0.1.2) Verify that the following functions are solutions of the given PDE.
a. Show that for any constant k, u( x, y) = ekx cos ky is a solution of Laplaces equation u xx + uyy = 0.
b. Show tha
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 8
18 November 2013
2
1. (Pinsky 3.2.4) Show that J1/2 ( x ) =
x sin x. (Hint: There at least two approaches here. You can use
the series denition or you can show that it satises the Bessel equation and has the right proper
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 10
Not handed in
1. (Pinsky 5.1.1516)
a. Let a > 0. Show that the Fourier transform of f ( ax ) is f(/ a)/ a.
b. Show that the Fourier transform of f ( x a) is eia f().
2. (Pinsky 5.1.23) If f ( x ), < x < is a complexvalu
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 9
25 November 2013
1. (Pinsky 4.2.15) Write down the associated Legendre functions P41 (s), P42 (s), P43 (s), and P44 (s).
2. (Pinsky 4.2.16) Show that Pkk (cos ) = ck (sin )k for k = 0, 1, 2, . . ., 0 , where ck is a suita
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 4
21 October 2013
1. (Pinsky 1.6.13) With reference to the generalized SturmLiouville problem
s( x ) ( x ) + [( x ) q( x )] ( x ) = 0,
cos ( a) L sin ( a) = 0,
cos (b) + L sin (b) = 0,
let L be the linear differential opera
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 7
11 November 2013
1. (Pinsky 2.5.14) Find the separated solutions of the wave equation utt = c2 (u xx + uyy ) in the square 0 <
x < L, 0 < y < L with the boundary conditions u x (t, 0, y) = u x (t, L, y) = 0 and uy (t, x,
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 3
14 October 2013
1. (Pinsky 1.2.18) Let f ( x ) = e x , < x < . What is the sum of the Fourier series for x = , x = ?
2. (Pinsky 1.2.19, 1.2.20)
a. Let f ( x ) and g( x ) be piecewise smooth functions for a < x < b. Show t
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 2
7 October 2013
1. (Pinsky 1.1.2, 1.1.4, 1.1.7) Compute the Fourier series of the following functions:
a. f ( x ) = x3 on the interval L < x < L
b. f ( x ) = e x on the interval L < x < L
c. f ( x ) =
0
1
L < x < 0
on the
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 5
28 October 2013
1. (Pinsky 2.2.6) Solve the initialvalue problem ut = Kuzz for t > 0 and 0 < z < L, with the boundary
conditions uz (t, 0) = uz (t, L) = 0 and the initial condition
u(0, z) = 3 + 4 cos
3 z
z
+ 7 cos
.
L
L
Fourier analysis and boundary value problems for ISP
MATH 381

Fall 2013
Math 381: Fall 2013
Problem Set 6
4 November 2013
1. (Pinsky 2.5.6) Solve Laplaces equation 2 u = 0 in the column 0 < x < L1 , 0 < y < L2 with boundary
conditions u x (0, y) = 0, u x ( L1 , y) = 0, u( x, 0) = T1 , u( x, L2 ) = T2 , where T1 and T2 are con