Legendre Polynomials Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Overview Solving the Legendre Equation Application Legendre Polynomials Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application Why are Legendre Polynomials Important? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application Why are Legendre Polynomials Important? 1. The generalized Legendre equation m 2 � � � 1 − x 2 � y ′′ − 2 xy ′ + λ − y = 0 arises when the 1 − x 2 equation ∆ u = f ( ρ ) u is solved with separation of variables in spherical coordinates. (QM: hydrogen atom!) The � � cos ( φ ) function y describes the polar part of the solution of ∆ u = f ( ρ ) u . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application Why are Legendre Polynomials Important? 1. The generalized Legendre equation m 2 � � � 1 − x 2 � y ′′ − 2 xy ′ + λ − y = 0 arises when the 1 − x 2 equation ∆ u = f ( ρ ) u is solved with separation of variables in spherical coordinates. (QM: hydrogen atom!) The � � cos ( φ ) function y describes the polar part of the solution of ∆ u = f ( ρ ) u . � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 is the 2. The Legendre equation special case with m = 0, which turns out to be the key to the generalized Legendre equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application Why are Legendre Polynomials Important? 1. The generalized Legendre equation m 2 � � � 1 − x 2 � y ′′ − 2 xy ′ + λ − y = 0 arises when the 1 − x 2 equation ∆ u = f ( ρ ) u is solved with separation of variables in spherical coordinates. (QM: hydrogen atom!) The � � cos ( φ ) function y describes the polar part of the solution of ∆ u = f ( ρ ) u . � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 is the 2. The Legendre equation special case with m = 0, which turns out to be the key to the generalized Legendre equation. 3. The solutions of both equations must be finite on [ − 1 , 1 ] . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application Why are Legendre Polynomials Important? 1. The generalized Legendre equation m 2 � � � 1 − x 2 � y ′′ − 2 xy ′ + λ − y = 0 arises when the 1 − x 2 equation ∆ u = f ( ρ ) u is solved with separation of variables in spherical coordinates. (QM: hydrogen atom!) The � � cos ( φ ) function y describes the polar part of the solution of ∆ u = f ( ρ ) u . � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 is the 2. The Legendre equation special case with m = 0, which turns out to be the key to the generalized Legendre equation. 3. The solutions of both equations must be finite on [ − 1 , 1 ] . 4. Because 0 is an ordinary point of the equation, it is natural to attempt a series solution. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ � c n n ( n − 1 ) x n − 2 ∑ n = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 ∑ ∑ n = 2 n = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n n = 2 n = 1 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ c n n ( n − 1 ) x n − 2 ∑ n = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ ∞ c n n ( n − 1 ) x n − 2 − c n n ( n − 1 ) x n ∑ ∑ n = 2 n = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ ∞ ∞ c n n ( n − 1 ) x n − 2 − c n n ( n − 1 ) x n − 2 c n nx n ∑ ∑ ∑ n = 2 n = 2 n = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ ∞ ∞ ∞ c n n ( n − 1 ) x n − 2 − c n n ( n − 1 ) x n − 2 c n nx n + λ c n x n ∑ ∑ ∑ ∑ n = 2 n = 2 n = 1 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ ∞ ∞ ∞ λ c n x n = c n n ( n − 1 ) x n − 2 − c n n ( n − 1 ) x n − 2 c n nx n + ∑ ∑ ∑ ∑ 0 n = 2 n = 2 n = 1 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ ∞ ∞ ∞ λ c n x n = c n n ( n − 1 ) x n − 2 − c n n ( n − 1 ) x n − 2 c n nx n + ∑ ∑ ∑ ∑ 0 n = 2 n = 2 n = 1 n = 0 ∞ ∑ c k + 2 ( k + 2 )( k + 1 ) x k k = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials

Overview Solving the Legendre Equation Application y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � � Series Solution of � 1 − x 2 � y ′′ − 2 xy ′ + λ y = 0 1 − x 2 � ∞ ∞ ∞ � c n n ( n − 1 ) x n − 2 − 2 x c n nx n − 1 + λ ∑ ∑ ∑ c n x n = 0 n = 2 n = 1 n = 0 ∞ ∞ ∞ ∞ λ c n x n = c n n ( n − 1 ) x n − 2 − c n n ( n − 1 ) x n − 2 c n nx n + ∑ ∑ ∑ ∑ 0 n = 2 n = 2 n = 1 n = 0 ∞ ∞ ∑ c k + 2 ( k + 2 )( k + 1 ) x k − ∑ c k k ( k − 1 ) x k k = 0 k = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Legendre Polynomials