1. (30 points) A spherical object of radius a is immersed in a zero temperature heat bath.
The sphere is heated with a -function source at r = r0 . The steady state diusion
equation for the temperature of the sphere i
University of Maryland
Department of Physics
Methods of Mathematical Physics
Dr. James F. Drake
A. V. Williams Bldg. (3311)
Oce Hours: by appointment o
1. (40 pts) The modied spherical Bessel equation is given by
z 2 y + 2zy [z 2 + n(n + 1)]y = 0
(a) Find the lowest order behavior of the solutions as z 0. Show that the
solutions are linearly independent. Identify
1. Arfken Chapter 6 : 1.10(a), 1.15(a)and (d), 2.1(a), 2.2, 2.3, 2.8, 7.1(a), 7.3(a)
2. Dene a cut in the complex z plane to make z 1/3 single valued. Evaluate
Arg [(i)1/3 ].
3. Dene cuts in the complex z plane to
1. Arfken Chapter 6 : 3.3, 4.1, 4.3, 4.4
2. If a function f (z ) is analytic on and within a closed contour C show that unless
it is a constant it takes on its maximum value on C .
Hint: Assume that f (z ) takes i
1. Arfken Chapter 6 : 5.1, 5.10, 5.11
Hint: for 5.11 (b), let z = rei and deform the t integral so that as changes
the t integral always converges this is the analytic continuation of the integral
1. Arfken Chapter 7 : 3.4, 3.5
2. Evaluate the following integral
for large x with x real and positive. You will nd that the lowest order term
scales as x1/2 and the next order as eix /x. Calculate
1. The Coulomb Wave equation is given by
2 L(L + 1)
(a) Classify the singular points of this equation.
(b) Find the behavior of the solutions for z very close to zero. Show that the
1. Arfken 9.6.11
2. Bessels equation is given by
+ z + (z 2 2 )y = 0.
Convert this equation to the standard form
+ k 2 (z )y = 0
and calculate the WKB solutions for large |z |. Where a
1. The inhomogeneous Airy equation is given by
xy = 1
This equation arises in mode conversion problems in plasma where an electromagnetic wave drives an electrostatic wave at its cuto. The 1 on the
This assignment is due on Thursday Dec. 1.
1. Arfken Chapter 10 : 1.2, 2.5
2. Expand the square-wave function f (x) [where f = 1 for x (1, 0) and f = 1
for x (0, 1)] in a series of Legendre polynomials over the i