Solutions to Laplace's Equation: Green's Function and Special Functions - Prof. James N. F, Assignments of Physics

Download Solutions to Laplace's Equation: Green's Function and Special Functions - Prof. James N. F and more Assignments Physics in PDF only on Docsity! PHY 6346 Fall 2008 Homework #7, Due Wednesday, October 29 1. Separating variables we showed that solutions to Laplace’s equation are superpositions of Φ = Jm(kρ) e imφe±kz. On an infinite volume, there is no quantization condition that gives a discrete set of k’s, and so the general solution is an integral over k. (a) Starting from Bessel’s equation, show that for ν > −12∫ ∞ 0 ρ dρ Jν(kρ) Jν(k ′ρ) = 1 k δ(k − k′). (b) Show that the Green’s function for free space is 1 |x − x′| = ∞∑ m=−∞ ∫ ∞ 0 dk Jm(kρ)Jm(kρ ′) eim(φ−φ′) e−k(z>−z<). 2. Many useful expressions involving special functions are found not by intent but acciden- tally, while computing something else. (a) Use appropriate special cases of the Green’s function in Problem 1 to show that ∫ ∞ 0 dk J0(kρ) e −k|z| = 1√ ρ2 + z2 ∞∑ m=−∞ Jm(kρ) Jm(kρ ′) eim(φ−φ′) = J0[k √ ρ2 + ρ′2 − 2ρρ′ cos(φ − φ′) ] ∞∑ m=−∞ im Jm(kρ) e imφ = eikρ cosφ Do not use the words or concept of “generating function” in obtaining your answer. (b) Use appropriate special cases of (a) to show [these should be very easy]: [J0(x)] 2 + 2 ∞∑ k=1 [Jk(x)] 2 = 1 J0(x) + 2 ∞∑ k=1 (−1)kJ2k(x) = cos(x), 2 ∞∑ k=0 (−1)kJ2k+1(x) = sin(x) (c) Obtain the integral representation Jm(x) = 1 im ∫ 2π 0 dφ 2π eix cosφ−imφ .