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Transcript of Legendre Polynomials

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Overview Solving the Legendre Equation Application

Legendre Polynomials

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Why are Legendre Polynomials Important?

1. The generalized Legendre equation(1 x2

)y2xy +

( m

2

1 x2

)y = 0 arises when the

equation u = f ()u is solved with separation of variablesin spherical coordinates. (QM: hydrogen atom!) Thefunction y

(cos()

)describes the polar part of the solution

of u = f ()u.2. The Legendre equation

(1 x2

)y2xy +y = 0 is the

special case with m = 0, which turns out to be the key tothe 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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Why are Legendre Polynomials Important?1. The generalized Legendre equation(

1 x2)

y2xy +(

m2

1 x2

)y = 0 arises when the

equation u = f ()u is solved with separation of variablesin spherical coordinates. (QM: hydrogen atom!) Thefunction y

(cos()

)describes the polar part of the solution

of u = f ()u.

2. The Legendre equation(1 x2

)y2xy +y = 0 is the

special case with m = 0, which turns out to be the key tothe 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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Why are Legendre Polynomials Important?1. The generalized Legendre equation(

1 x2)

y2xy +(

m2

1 x2

)y = 0 arises when the

equation u = f ()u is solved with separation of variablesin spherical coordinates. (QM: hydrogen atom!) Thefunction y

(cos()

)describes the polar part of the solution

of u = f ()u.2. The Legendre equation

(1 x2

)y2xy +y = 0 is the

special case with m = 0, which turns out to be the key tothe 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.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Why are Legendre Polynomials Important?1. The generalized Legendre equation(

1 x2)

y2xy +(

m2

1 x2

)y = 0 arises when the

equation u = f ()u is solved with separation of variablesin spherical coordinates. (QM: hydrogen atom!) Thefunction y

(cos()

)describes the polar part of the solution

of u = f ()u.2. The Legendre equation

(1 x2

)y2xy +y = 0 is the

special case with m = 0, which turns out to be the key tothe 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 naturalto attempt a series solution.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Why are Legendre Polynomials Important?1. The generalized Legendre equation(

1 x2)

y2xy +(

m2

1 x2

)y = 0 arises when the

equation u = f ()u is solved with separation of variablesin spherical coordinates. (QM: hydrogen atom!) Thefunction y

(cos()

)describes the polar part of the solution

of u = f ()u.2. The Legendre equation

(1 x2

)y2xy +y = 0 is the

special case with m = 0, which turns out to be the key tothe 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.Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn22x

n=1

cnnxn1 +

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0

(1 x2

) n=2

cnn(n1)xn22x

n=1

cnnxn1 +

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn2

2x

n=1

cnnxn1 +

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn22x

n=1

cnnxn1

+

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn22x

n=1

cnnxn1 +

n=0

cnxn

= 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn22x

n=1

cnnxn1 +

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn22x

n=1

cnnxn1 +

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1

Overview Solving the Legendre Equation Application

Series Solution of(1 x2

)y2xy+y = 0

(1 x2

)y2xy +y = 0(

1 x2)

n=2

cnn(n1)xn22x

n=1

cnnxn1 +

n=0

cnxn = 0

n=2

cnn(n1)xn2

n=2

cnn(n1)xn

n=1

2cnnxn+

n=0

cnxn = 0

k=0

ck+2(k +2)(k +1)xk

k=2

ckk(k1)xk

k=1

2ckkxk+

k=0

ckxk = 0

2c2 +c0 +6c3x2c1x+c1x+

k=2

[(k +2)(k +1)ck+2 k(k1)ck 2kck +ck

]xk = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Legendre Polynomials

• logo1