Answer Set 5
Physics 204B
0
1
1
2
Riley 13.1 g ( ) = 1 [ 0 et eit dt + et eit dt] = 1 [ 1+i + 1i ] = 1 1+2 . Since the
2
2
2
function f (t) is even, you could say straight o that the sine part of the Fourier transform
must vanish and the cosine contributi
Problem Set 9
Physics 204B
Due Friday March 14, 2014; late HW accepted until 1 PM on Monday March 17
For grading purposes, Set 9 will count as 0.75 problem sets.
1. a) Given measurements of x (98, 101, 102, 100, 99) and of y (21.2, 20.8, 18.1, 20.3, 19.6,
Answer Set 1
Physics 204B
1. a) u/x = 2x 2 = v/y implies v (x, y ) = 2xy 2y + terms independent of y . Also,
u/y = 4y = v/x implies v (x, y ) = 4xy + terms independent of x. These two
expressions for v are incompatible, so no analytic function w exists.
b
Problem Set 8, Physics 204B
Due Monday March 10, 2014 Late HW accepted until class Wednesday March 12 Do Riley 30.24 and 30.26, plus the following: 1. a) Plot the probability of x 5's in n tosses of a die as a function of x, for n = 6, 12, 60, and 120. b)
Problem Set 7
Physics 204B
Due Monday March 3, 2014; late HW accepted until Friday March 7
Reminder: no class March 5, but well start at 11:45 on February 28 and March 7
Do Riley 30.10 and 30.12, plus the following:
1. A candy vending machine is broken: w
Problem Set 5 Physics 204B
Due in class Monday February 10, 2014; late homework accepted until class on February 12 Do Riley 13.1 and 13.4, plus the following: 1. a) If a function f (x) has a Fourier transform g(k) that falls off as k13 as |k| , how df qu
Problem Set 6
Physics 204B
Due in class Monday February 24, 2014; late homework accepted until class February 26
(Counts as 1.5 homework assignments)
2
1. The one-dimensional neutron diusion equation with a plane source at x = 0 is D d dx(2x) +
K 2 D(x) =
Answer Set 2
Physics 204B
1. a) Singularities of order 1 at z = 0, z = 1, z = 1. Residues are -1, e/2, 1/2e, respectively.
b) Essential singularity at z = 0, with residue 1/2. Note that the Taylor expansion of ew converges
for any nite, non-zero w, so you
Answer Set 3 Physics 204B
1. a)
cos xdx 0 1+x2 +x4 e dz 1 = 2 Re 1+z 2 +z 4 . Poles are at z 2 = - 1 23 i (from the quadratic formula, since 2 the denominator is a quadratic polynomial in z 2 ). These lie on the unit circle, at angles 2/3 from the positi
Midterm Answers
Physics 204B, 2013
1
1
1. Take g (t) = 1/t and f (t) = at t. Then f (t) = a 2t and f (t) = 4t3/2 . The saddle
1
1
point is at the zero of f , t = 4a2 . At the saddle, f ( 4a2 ) = 2a3 . This is real, so
1
= 2 2 arg(f ) = 2 . Thats a good d
Midterm
Physics 204B, 2013
The three questions have equal weights. Do them all. A possibly useful formula:
ei g (z0 )esf (z0 ) s|f 2(z0 )|
c+i 1 s(at t)
1
dt
2i ci t e
1. Use the method of steepest descents to approximate
for large
s, where a, c, s are po
Answer Set 8
Physics 204B
Riley 30.10 a) After getting the same answer to two questions, here are the possibilities, along
with the probabilities that the traveller would expect in advance for each outcome.
9
Ascii 11 TT 11 16
16
16
FF
5
16
11 1
16 16
5
1
Answer Set 6, Physics 204B
1. The charge distribution is the inverse transform of F , (r) =
1
(2 )3
eikr
2
1+ k2
d3 k . Choose k -space
a
coordinates so that kz is aligned with r. This means your k -space coordinates will be dierent for dierent values of
Answer Set 4 Physics 204B
Riley 24.21 Consider
(ln z)2 dz 1+z 2
over the path specified. The only singularity inside the contour
2 3
is at z = i, so the integral becomes 2i (ln i) = - 4 . On the large semicircle, parametrize 2i R)2 with z = Rei . The inte
Problem Set 1
Physics 204B
Due Monday January 13, 2014
Late HW accepted until class January 15
1. For each of the following functions u(x, y ), state whether an analytic function w(z ) exists
with w(z ) = u(x, y ) + iv (x, y ) and z = x + iy . If so, nd v