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Special Functions

Source lecture(s): PHY621 Lec5

Intuition

Bessel, Legendre, and other named functions recur so often in physics that they deserve catalog entries.

Formal Definition

Bessel \(J_n(x)\) solves \(x^2y''+xy'+(x^2-n^2)y=0\). Legendre \(P_l(x)\) solves \((1-x^2)y''-2xy'+l(l+1)y=0\).

Mathematical Formulation

\[J_n(x)=\sum_{m=0}^\infty \frac{(-1)^m}{m!(m+n)!}\left(\frac{x}{2}\right)^{2m+n}\]

Derivation

Use Frobenius series \(y=\sum a_m x^{m+r}\); equate lowest-power terms to find \(r\) and recurrences.

Worked Example

Small-\(x\) behavior: \(J_0(x)\approx1-\frac{x^2}{4}\).

Common Mistakes

  • Using Bessel \(Y_n\) without checking singular behavior at \(x=0\).
  • Using a weight function for Legendre orthogonality — \(P_l\) are orthogonal on \([-1,1]\) with weight \(1\) (the weight \((1-x^2)^{-1/2}\) belongs to Chebyshev polynomials).

Quiz

Q1. What ODE defines Bessel functions of order \(n\)?

Answer

\(x^2y''+xy'+(x^2-n^2)y=0\).