HOMEWORK PROBLEMS 8 PHYS 516, Fall 2014 Special Functions and Partial Differential Equations Due date: Monday, 11/24/14 — 3.00pm (i.e. in class) 1. (a) Arfken & Weber Problems 11.7.8 and 12.4.10. You may need to read the start of Section 11.7 to remind yourself about the definition of spherical Bessel functions. (b) Then use your result from 12.4.10 to prove the integral identity s In+1/2 (z) = z 2π Z 1 −1 ezx Pn (x) dx (1) for the modified Bessel function. [24 points credit] 2. Prove Laplace’s useful integral representation of the Legendre function of the first kind Pn (z) Z in √ 1 πh Pn (z) = z + z 2 − 1 cos φ dφ , |z| > 1 , (2) π 0 by showing that it satisfies the generating function relation √ ∞ X 1 = tn Pn (z) , 2 1 − 2tz + t n=0 |t| < 1 . (3) Hint: you will need to perform a contour integration, the form of which you have seen before. [15 points credit] 1 3. Arfken & Weber Problems 13.1.12 and 13.1.18. For 13.1.18, use instead the function f (x) = x6 to determine the coefficients an , and then check your results using Table 13.1. [25 points credit] 4. Arfken & Weber Problems 9.3.6 & 9.3.7. Then identify appropriate eigenvalues (or quantum numbers) to express the separation constant, so that the general solution can be written as an infinite series of well-known special functions. If the boundary condition for the problem is that the temperature is exactly zero at the outer radius R of the cylinder, discuss how this constrains the form of the solution. [16 points credit] 5. (a) Using the technique of separation of variables, show that the solution to the three-dimensional diffusion equation for temperature T , in the special case of cylindrical symmetry about an axis r = 0 , assumes the form (for µ to be specified) T (r, t) = rµ Z ∞ 0 dk f (k) J0 (k r) e−k 2 κt , (4) where f (k) is an arbitrary function of the dimensionless integration variable k , κ is the diffusion coefficient, and J0 (z) is the ordinary Bessel function of order zero. (b) Using a familiar integral representation of the ordinary Bessel functions, prove the Laplace transform identity Z ∞ 0 1 e−αk J0 (k r) dk = √ 2 α + r2 . Hint: Part of the analysis of Problem 2 should come in handy. 2 (5) (c) Consider cases where f (k) ∝ k . Use appropriate derivatives of the Laplace transform identity to demonstrate that such cases are appropriate for the diffusion problem where the heat (i.e. temperature) is initially (i.e. at t = 0 ) infinitely concentrated at r = 0 , and zero elsewhere. Can you determine a simple analytic form for T (r, t) for such scenarios? [20 points credit] 3