1. 4.4 Legendre Functions 4.4.1. Legendre polynomials. The differential equation: = -1 < x < 1 around x = 0

2. Radii of Convergence Thus R = 1 so each series converges in -1 < x < 1 Example: Steady state temperature distribution within a sphere subjected to a known temperature distribution on its surface. Solutions bounded on -1 < x <1 F(x) bounded on an interval I, means exists M / F(x)| ≤ M for all x in I Sometimes series terminate; it is bounded on the interval; a finite degree Polynomial! Specifically, if λ = n(n + 1)

3. Normalization of polynomials: P n (1) = 1

4. Rodrigues's formula Legendre polynomials (scale to P n (1) = 1) are the solution of the Legendre equation Orthogonality (1-x 2 )y’’ – 2xy’ = [(1-x 2 )y’]’

5. P j, P k are solutions of

6. 4.4.3. Generating function and properties generating function Symmetry F(x) = F(-x) even function F(x) = -F(-x) odd function

7. Taking partial derivative w/r to r Finally the case, i = j = n Taking partial derivative w/r to x

8. Closure property The Legendre polynomials form a closed set or basis, which can be used to represent any other functionunit vectors îĵ In the same way any vector in 3D can be expanded using three orthogonal vectors i, j, k, which form a finite basis set In the same way the Legendre polynomials for an infinite orthogonal basis set, defined for x in the interval –1 ≤ x ≤ +1 P 0 (x), P 1 (x), P 2 (x), P 3 (x), …

9. 4.5 Singular Integrals; Gamma Function Integrals: 1) singular (or improper): if one or both integration limits are infinite and/or if the integrand is unbounded on the interval; 2) otherwise, it is regular (or proper). I is convergent if the limit exists; otherwise, I is divergent Converges for p > 1 and diverges for p ≤ 1

11. Analogous to a p-series, the horizontal p-integral