Legendre多项式的生成函数 √1-2at+t2 ∑P()H<2 l=0 可以推出许多有用的结果例如令=1,得到 P(1)=
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials Legendreõª)¤¼ê 1 √ 1−2xt+t 2 = X ∞ l=0 Pl(x)t l |t| < x± p x 2−1 ±íÑNõk^(J©~X-x = 1§ íØ1 1 √ 1 − 2t + t 2 = 1 1 − t X ∞ l=0 Pl(1)t l = X ∞ l=0 t l =⇒ Pl(1) = 1 C. S. Wu 1Ôù ¥¼ê()
Legendre多项式的生成函数 √1-2at+t2 ∑P()H<2 l=0 可以推出许多有用的结果.例如令x=1,得到 推论1 2t+t2 →>P(1) ∑P(1)2=∑ l=0 l=0
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials Legendreõª)¤¼ê 1 √ 1−2xt+t 2 = X ∞ l=0 Pl(x)t l |t| < x± p x 2−1 ±íÑNõk^(J©~X-x = 1§ íØ1 1 √ 1 − 2t + t 2 = 1 1 − t X ∞ l=0 Pl(1)t l = X ∞ l=0 t l =⇒ Pl(1) = 1 C. S. Wu 1Ôù ¥¼ê()
Legendre多项式的生成函数 v1-2t+12 ∑P()2|<|v2-1 可以推出许多有用的结果因为 1-2(+121-2(-x)(-1)+(-1)2 P(x)=>P(=)( (-x)=(-)P(x)
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials Legendreõª)¤¼ê 1 √ 1−2xt+t 2 = X ∞ l=0 Pl(x)t l |t| < x± p x 2−1 ±íÑNõk^(J©Ï 1 √ 1−2xt+t 2 = 1 p 1−2(−x)(−t)+(−t) 2 íØ2 X ∞ l=0 Pl(x)t l = X ∞ l=0 Pl(−x)(−t) l ⇒ Pl(−x)= (−) lPl(x) C. S. Wu 1Ôù ¥¼ê()
Legendre多项式的生成函数 v1-2t+12 ∑P()2|<|v2-1 可以推出许多有用的结果.因为 1-2xt+t2 √1-2(-)(-t)+(-t)2 论 ∑P(a)t=∑P(-a)(-1)→P(-)=(-)P(x)
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials Legendreõª)¤¼ê 1 √ 1−2xt+t 2 = X ∞ l=0 Pl(x)t l |t| < x± p x 2−1 ±íÑNõk^(J©Ï 1 √ 1−2xt+t 2 = 1 p 1−2(−x)(−t)+(−t) 2 íØ2 X ∞ l=0 Pl(x)t l = X ∞ l=0 Pl(−x)(−t) l ⇒ Pl(−x)= (−) lPl(x) C. S. Wu 1Ôù ¥¼ê()
Legendre多项式的生成函数 2 xt +t2 ∑P(n)<士 l=0 可以推出许多有用的结果直接在=0点 作 Lawlor展开,就得到 PI)
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials Legendreõª)¤¼ê 1 √ 1−2xt+t 2 = X ∞ l=0 Pl(x)t l |t| < x± p x 2−1 ±íÑNõk^(J©3t = 0: TaylorÐm§Ò íØ3 Pl(x) = X [l/2] r=0 (−) r (2l − 2r)! 2 lr!(l − r)!(l − 2r)!x l−2r C. S. Wu 1Ôù ¥¼ê()