Separation of Variables -- Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables – Legendre Equations

Bernd Schroder¨

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is a solution of the form u = R(ρ)T(θ)P(φ). 3. The special form of this solution function allows us to replace the original partial differential equation with several ordinary differential equations. 4. Key step: If f (ρ) = g(θ,φ), then f and g must be constant. 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations. 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables 1. Solution technique for partial differential equations.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 3. The special form of this solution function allows us to replace the original partial differential equation with several ordinary differential equations. 4. Key step: If f (ρ) = g(θ,φ), then f and g must be constant. 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations. 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 4. Key step: If f (ρ) = g(θ,φ), then f and g must be constant. 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations. 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations. 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is a solution of the form u = R(ρ)T(θ)P(φ). 3. The special form of this solution function allows us to replace the original partial differential equation with several ordinary differential equations. 4. Key step: If f (ρ) = g(θ,φ), then f and g must be constant. 5. Solutions of the ordinary differential equations we obtain must typically be processed some more to give useful results for the partial differential equations.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations plus about 200 pages of really awesome functional analysis.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

How Deep?

plus about 200 pages of really awesome functional analysis. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 1. For constant f , this is an eigenvalue equation for the Laplace operator, which arises, for example, in separation of variables for the heat equation or the wave equation. 2. The time independent Schrodinger¨ equation h¯ − ∆φ + Vφ = Eφ describes certain quantum 2m mechanical systems, for example, the electron in a h hydrogen atom. m is the mass of the electron, h¯ = , 2π where h is Planck’s constant, V(ρ) is the electric potential and E is the energy eigenvalue. 3. The equation ∆u = f (ρ)u had already been investigated in electrodynamics when its importance for the states of an electron in a hydrogen atom became clear.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

The Equation ∆u = f (ρ)u

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 2. The time independent Schrodinger¨ equation h¯ − ∆φ + Vφ = Eφ describes certain quantum 2m mechanical systems, for example, the electron in a h hydrogen atom. m is the mass of the electron, h¯ = , 2π where h is Planck’s constant, V(ρ) is the electric potential and E is the energy eigenvalue. 3. The equation ∆u = f (ρ)u had already been investigated in electrodynamics when its importance for the states of an electron in a hydrogen atom became clear.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 3. The equation ∆u = f (ρ)u had already been investigated in electrodynamics when its importance for the states of an electron in a hydrogen atom became clear.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

The Equation ∆u = f (ρ)u 1. For constant f , this is an eigenvalue equation for the Laplace operator, which arises, for example, in separation of variables for the heat equation or the wave equation. 2. The time independent Schrodinger¨ equation h¯ − ∆φ + Vφ = Eφ describes certain quantum 2m mechanical systems, for example, the electron in a h hydrogen atom. m is the mass of the electron, h¯ = , 2π where h is Planck’s constant, V(ρ) is the electric potential and E is the energy eigenvalue.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 2 1 cos(φ) 1 R00TP + R0TP + RTP00 + RTP0 + RT00P = f (ρ)RTP ρ ρ2 ρ2 sin(φ) ρ2 sin2(φ) R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R Both sides must be constant. R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. (QM: Laguerre polys.)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 2 1 cos(φ) 1 R00TP + R0TP + RTP00 + RTP0 + RT00P = f (ρ)RTP ρ ρ2 ρ2 sin(φ) ρ2 sin2(φ) R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 2 1 cos(φ) 1 + R0TP + RTP00 + RTP0 + RT00P = f (ρ)RTP ρ ρ2 ρ2 sin(φ) ρ2 sin2(φ) R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 R00TP

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 1 cos(φ) 1 + RTP00 + RTP0 + RT00P = f (ρ)RTP ρ2 ρ2 sin(φ) ρ2 sin2(φ) R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 2 R00TP + R0TP ρ

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations cos(φ) 1 + RTP0 + RT00P = f (ρ)RTP ρ2 sin(φ) ρ2 sin2(φ) R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 2 1 R00TP + R0TP + RTP00 ρ ρ2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 1 + RT00P = f (ρ)RTP ρ2 sin2(φ) R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 2 1 cos(φ) R00TP + R0TP + RTP00 + RTP0 ρ ρ2 ρ2 sin(φ)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations = f (ρ)RTP

R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 2 1 cos(φ) 1 R00TP + R0TP + RTP00 + RTP0 + RT00P ρ ρ2 ρ2 sin(φ) ρ2 sin2(φ)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations R00 R0 P00 cos(φ) P0 1 T00 ρ2 + 2ρ + + + = ρ2f (ρ) R R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations R0 P00 cos(φ) P0 1 T00 + 2ρ + + + = ρ2f (ρ) R P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations P00 cos(φ) P0 1 T00 + + + = ρ2f (ρ) P sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part) ∂ 2u 2 ∂u 1 ∂ 2u cos(φ) ∂u 1 ∂ 2u + + + + = f (ρ)u ∂ρ2 ρ ∂ρ ρ2 ∂φ 2 ρ2 sin(φ) ∂φ ρ2 sin2(φ) ∂θ 2 2 1 cos(φ) 1 R00TP + R0TP + RTP00 + RTP0 + RT00P = f (ρ)RTP ρ ρ2 ρ2 sin(φ) ρ2 sin2(φ) R00 R0 ρ2 + 2ρ R R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations cos(φ) P0 1 T00 + + = ρ2f (ρ) sin(φ) P sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 1 T00 + = ρ2f (ρ) sin2(φ) T Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations = ρ2f (ρ)

Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Bring all terms that depend on ρ to the right side:

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R Both sides must be constant. R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. (QM: Laguerre polys.)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Both sides must be constant. R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. (QM: Laguerre polys.)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. (QM: Laguerre polys.)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R Both sides must be constant.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. (QM: Laguerre polys.)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R Both sides must be constant. R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations (QM: Laguerre polys.)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R Both sides must be constant. R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

P00 cos(φ) P0 1 T00 R00 R0 + + = ρ2f (ρ) − ρ2 − 2ρ , P sin(φ) P sin2(φ) T R R Both sides must be constant. R00 R0 ρ2f (ρ) − ρ2 − 2ρ = −λ, or R R ρ2R00 + 2ρR0 − λR + ρ2f (ρ)R = 0. (QM: Laguerre polys.) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations P00 cos(φ) P0 1 T00 + + = −λ P sin(φ) P sin2(φ) T P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + = −λ sin2(φ) P P T P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = − P P T Both sides must be constant. T00 − = c leads to T00 + cT = 0. T But T must be 2π-periodic. Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + = −λ sin2(φ) P P T P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = − P P T Both sides must be constant. T00 − = c leads to T00 + cT = 0. T But T must be 2π-periodic. Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part) P00 cos(φ) P0 1 T00 + + = −λ P sin(φ) P sin2(φ) T

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = − P P T Both sides must be constant. T00 − = c leads to T00 + cT = 0. T But T must be 2π-periodic. Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part) P00 cos(φ) P0 1 T00 + + = −λ P sin(φ) P sin2(φ) T P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + = −λ sin2(φ) P P T

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Both sides must be constant. T00 − = c leads to T00 + cT = 0. T But T must be 2π-periodic. Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations T00 − = c leads to T00 + cT = 0. T But T must be 2π-periodic. Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations But T must be 2π-periodic. Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Thus c = m2, where m is a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part) P00 cos(φ) P0 1 T00 + + = −λ P sin(φ) P sin2(φ) T P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + = −λ sin2(φ) P P T P00 P0 T00 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = − P P T Both sides must be constant. T00 − = c leads to T00 + cT = 0. T But T must be 2π-periodic.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations So the function T must be of the form T(θ) = c1 cos(mθ) + c2 sin(mθ).

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations P00 P0 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = m2 P P sin2(φ)P00 + sin(φ)cos(φ)P0 + λ sin2(φ) − m2 P = 0 cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) This equation is complicated, because it involves trigonometric functions. It turns out that the substitution z = cos(φ) will simplify the equation.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations sin2(φ)P00 + sin(φ)cos(φ)P0 + λ sin2(φ) − m2 P = 0 cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) This equation is complicated, because it involves trigonometric functions. It turns out that the substitution z = cos(φ) will simplify the equation.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

P00 P0 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = m2 P P

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) This equation is complicated, because it involves trigonometric functions. It turns out that the substitution z = cos(φ) will simplify the equation.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

P00 P0 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = m2 P P sin2(φ)P00 + sin(φ)cos(φ)P0 + λ sin2(φ) − m2 P = 0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations This equation is complicated, because it involves trigonometric functions. It turns out that the substitution z = cos(φ) will simplify the equation.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

P00 P0 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = m2 P P sin2(φ)P00 + sin(φ)cos(φ)P0 + λ sin2(φ) − m2 P = 0 cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations It turns out that the substitution z = cos(φ) will simplify the equation.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

P00 P0 sin2(φ) + sin(φ)cos(φ) + λ sin2(φ) = m2 P P sin2(φ)P00 + sin(φ)cos(φ)P0 + λ sin2(φ) − m2 P = 0 cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) This equation is complicated, because it involves trigonometric functions.

Separating the Equation ∆u = f (ρ)u (Polar Part)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d d d d = P z = P cos(φ) dz dφ dz dφ d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d P dφ

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d d = P cos(φ) dz dφ d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d P = P z dφ dz dφ

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ d = P − sin(φ) dz

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d d = P − sin(φ) dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ d = P − sin(φ) dz d2 P dφ 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d2 d = sin2(φ) P − cos(φ) P dz2 dz

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution d d d d d P = P z = P cos(φ) dφ dz dφ dz dφ d = P − sin(φ) dz d2 d d P = P − sin(φ) dφ 2 dφ dz d d d = P − sin(φ) + P − cos(φ) dφ dz dz d d d = P − sin(φ) − sin(φ) + P − cos(φ) dz dz dz d2 d = sin2(φ) P − cos(φ) P dz2 dz logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP m2 1 − z2 − 2z + λ − P = 0 dz2 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP m2 1 − z2 − 2z + λ − P = 0 dz2 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations cos(φ) dP m2 + − sin(φ)+ λ − P = 0 sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP m2 1 − z2 − 2z + λ − P = 0 dz2 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP sin2(φ) −cos(φ) dz2 dz

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations m2 + λ − P = 0 sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP m2 1 − z2 − 2z + λ − P = 0 dz2 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP sin2(φ) −cos(φ) + − sin(φ) dz2 dz sin(φ) dz

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP m2 1 − z2 − 2z + λ − P = 0 dz2 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations d2P dP m2 1 − z2 − 2z + λ − P = 0 dz2 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations dP m2 − 2z + λ − P = 0 dz 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P 1 − z2 dz2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations m2 + λ − P = 0 1 − z2

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP 1 − z2 − 2z dz2 dz

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations = 0

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

cos(φ) m2 P00 + P0 + λ − P = 0 sin(φ) sin2(φ) d2P dP cos(φ) dP m2 sin2(φ) −cos(φ) + − sin(φ)+ λ − P = 0 dz2 dz sin(φ) dz sin2(φ) d2P dP m2 sin2(φ) − 2cos(φ) + λ − P = 0 dz2 dz sin2(φ) d2P dP m2 1 − z2 − 2z + λ − P dz2 dz 1 − z2

Generalized Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations For nonnegative integers l, the differential equation 1 − x2 y00 − 2xy0 + l(l + 1)y = 0 is called the Legendre equation. Formally, both are actually families of differential equations, because m,λ and l are parameters. m is a nonnegative integer, because this is required through the equation for T(θ) in the separation of variables.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre Equations Let λ be a real number and let m be a nonnegative integer. The differential equation m2 1 − x2 y00 − 2xy0 + λ − y = 0 1 − x2 is called the generalized Legendre equation.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations Formally, both are actually families of differential equations, because m,λ and l are parameters. m is a nonnegative integer, because this is required through the equation for T(θ) in the separation of variables.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations m is a nonnegative integer, because this is required through the equation for T(θ) in the separation of variables.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre Equations Let λ be a real number and let m be a nonnegative integer. The differential equation m2 1 − x2 y00 − 2xy0 + λ − y = 0 1 − x2 is called the generalized Legendre equation. For nonnegative integers l, the differential equation 1 − x2 y00 − 2xy0 + l(l + 1)y = 0 is called the Legendre equation. Formally, both are actually families of differential equations, because m,λ and l are parameters. m is a nonnegative integer, because this is required through the equation for T(θ) in the separation of variables. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations In the Legendre equation 1 − x2 y00 − 2xy0 + l(l + 1)y = 0, the parameter λ should be of the form l(l + 1) with l a nonnegative integer, because of the following: 1. For λ not of this form the solutions go to inﬁnity as z approaches ±1. 2. z approaching ±1 corresponds to cos(φ) approaching ±1, which corresponds to φ approaching 0 and π. 3. So, physically this would mean that for λ 6= l(l + 1), the function u would be inﬁnite on the z-axis, which is not sensible.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre Equations

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 1. For λ not of this form the solutions go to inﬁnity as z approaches ±1. 2. z approaching ±1 corresponds to cos(φ) approaching ±1, which corresponds to φ approaching 0 and π. 3. So, physically this would mean that for λ 6= l(l + 1), the function u would be inﬁnite on the z-axis, which is not sensible.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre Equations In the Legendre equation 1 − x2 y00 − 2xy0 + l(l + 1)y = 0, the parameter λ should be of the form l(l + 1) with l a nonnegative integer, because of the following:

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 2. z approaching ±1 corresponds to cos(φ) approaching ±1, which corresponds to φ approaching 0 and π. 3. So, physically this would mean that for λ 6= l(l + 1), the function u would be inﬁnite on the z-axis, which is not sensible.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations 3. So, physically this would mean that for λ 6= l(l + 1), the function u would be inﬁnite on the z-axis, which is not sensible.

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre Equations In the Legendre equation 1 − x2 y00 − 2xy0 + l(l + 1)y = 0, the parameter λ should be of the form l(l + 1) with l a nonnegative integer, because of the following: 1. For λ not of this form the solutions go to inﬁnity as z approaches ±1. 2. z approaching ±1 corresponds to cos(φ) approaching ±1, which corresponds to φ approaching 0 and π.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations