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4.4 Legendre Functions 4.4.1. Legendre polynomials. The differential equation:

4.4 Legendre Functions 4.4.1. Legendre polynomials. The differential equation:. =. -1 < x < 1. around x = 0. Radii of Convergence. Thus R = 1 so each series converges in -1 < x < 1 Example:

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4.4 Legendre Functions 4.4.1. Legendre polynomials. The differential equation:

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  1. 4.4 LegendreFunctions 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: Pn (1) = 1

  4. Legendre polynomials (scale to Pn (1) = 1) are the solution of the Legendre equation Rodrigues's formula Orthogonality (1-x2)y’’ – 2xy’ = [(1-x2)y’]’

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

  8. 4.5 Singular Integrals; Gamma Function 4.5.1. An integral is said to be singular (or improper) integral: if one or both integration limits are infinite and/or if the integrand is unbounded on the interval; otherwise, it is regular (or proper). Convergent: If the limit exists; if not, I is divergent Converges for p > 1 and diverges for p ≤ 1

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