Separation of Variables -- Legendre Equations - Louisiana Tech ...

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables – Legendre
Equations
Bernd Schröder
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Bernd Schröder 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
Separation of Variables
1. Solution technique for partial differential equations.
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Bernd Schröder 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
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).
logo1
Bernd Schröder 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
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.
logo1
Bernd Schröder 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
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.
logo1
Bernd Schröder 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
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 Schröder 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
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.
6. Some very powerful and deep theorems can be used to
formally justify the approach for many equations involving
the Laplace operator.
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Bernd Schröder 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
How Deep?
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Bernd Schröder 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
How Deep?
plus about 200 pages of really
awesome functional analysis.
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Bernd Schröder 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
The Equation ∆u = f (ρ)u
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Bernd Schröder 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
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.
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Bernd Schröder 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
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 Schrödinger equation
− ¯h
∆φ + Vφ = Eφ describes certain quantum
2m
mechanical systems, for example, the electron in a
hydrogen atom. m is the mass of the electron, ¯h = h
2π ,
where h is Planck’s constant, V(ρ) is the electric potential
and E is the energy eigenvalue.
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Bernd Schröder 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
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 Schrödinger equation
− ¯h
∆φ + Vφ = Eφ describes certain quantum
2m
mechanical systems, for example, the electron in a
hydrogen atom. m is the mass of the electron, ¯h = 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.
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u = f (ρ)u
∂θ 2
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
R ′′ TP
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u = f (ρ)u
∂θ 2
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u = f (ρ)u
∂θ 2
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ
2 RTP′′
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u = f (ρ)u
∂θ 2
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u = f (ρ)u
∂θ 2
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2
1
ρ
= f (ρ)u
2 sin 2 (φ) RT′′ P
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′
ρ
R
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′
ρ + 2ρ
R R
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
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Bernd Schröder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
P ′
P

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
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
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Bernd Schröder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
T ′′
T

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
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
T ′′
T = ρ2 f (ρ)
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
T ′′
T = ρ2 f (ρ)
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
P ′′
P
T ′′
T = ρ2 f (ρ)
cos(φ) P
+
sin(φ)
′ 1
+
P sin 2 T
(φ)
′′
T = ρ2 2 R′′ R′
f (ρ) − ρ − 2ρ
R R ,
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
P ′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P
sin 2 (φ)
Both sides must be constant.
T ′′
T = ρ2 f (ρ)
T ′′
T = ρ2 2 R′′ R′
f (ρ) − ρ − 2ρ
R R ,
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
P ′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P
sin 2 (φ)
Both sides must be constant.
ρ 2 2 R′′ R′
f (ρ) − ρ − 2ρ = −λ, or
R R
T ′′
T = ρ2 f (ρ)
T ′′
T = ρ2 2 R′′ R′
f (ρ) − ρ − 2ρ
R R ,
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
T ′′
T = ρ2 f (ρ)
P ′′ cos(φ) P
+
P sin(φ)
′ 1
+
P sin 2 T
(φ)
′′
T = ρ2 2 R′′ R′
f (ρ) − ρ − 2ρ
R R ,
Both sides must be constant.
ρ 2 2 R′′ R′
f (ρ) − ρ − 2ρ = −λ, or
R R
ρ 2 R ′′ + 2ρR ′ − λR + ρ 2 f (ρ) R = 0.
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u 2 ∂u 1
+ +
∂ρ 2 ρ ∂ρ ρ2 ∂ 2u cos(φ)
+
∂φ 2 ρ2 sin(φ)
R ′′ TP + 2
ρ R′ TP + 1
ρ 2 RTP′′ + cos(φ)
ρ 2 sin(φ) RTP′ +
2 R′′ R′ P′′
ρ + 2ρ +
R R P
∂u
∂φ +
1
ρ2 sin 2 ∂
(φ)
2u ∂θ 2 = f (ρ)u
1
ρ2 sin 2 (φ) RT′′ P = f (ρ)RTP
+ cos(φ)
sin(φ)
P ′ 1
+
P sin2 (φ)
Bring all terms that depend on ρ to the right side:
T ′′
T = ρ2 f (ρ)
P ′′ cos(φ) P
+
P sin(φ)
′ 1
+
P sin 2 T
(φ)
′′
T = ρ2 2 R′′ R′
f (ρ) − ρ − 2ρ
R R ,
Both sides must be constant.
ρ 2 2 R′′ R′
f (ρ) − ρ − 2ρ = −λ, or
R R
ρ 2 R ′′ + 2ρR ′ − λR + ρ 2 f (ρ) R = 0. (QM: Laguerre polys.)
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
P ′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
sin 2 (φ) P′′
P
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
+ sin(φ)cos(φ)P′
P + λ sin2 (φ) = − T′′
T
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
sin 2 (φ) P′′
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
+ sin(φ)cos(φ)P′
P
Both sides must be constant.
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
P + λ sin2 (φ) = − T′′
T
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = − T′′
Both sides must be constant.
−
T
T′′
T = c leads to T′′ + cT = 0.
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = − T′′
Both sides must be constant.
−
T
T′′
T = c leads to T′′ + cT = 0.
But T must be 2π-periodic.
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = − T′′
T
Both sides must be constant.
− T′′
T = c leads to T′′ + cT = 0.
But T must be 2π-periodic. Thus c = m2 , where m is a
nonnegative integer.
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Azimuthal Part)
P ′′
P
sin 2 (φ) P′′
P
cos(φ) P
+
sin(φ)
′ 1
+
P sin2 T
(φ)
′′
T
= −λ
T′′
+ sin(φ)cos(φ)P′ +
P T = −λ sin2 (φ)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = − T′′
T
Both sides must be constant.
− T′′
T = c leads to T′′ + cT = 0.
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).
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Polar Part)
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin 2 (φ) P′′
P
+ sin(φ)cos(φ)P′
P + λ sin2 (φ) = m 2
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin 2 (φ) P′′
P
sin 2 (φ)P ′′ + sin(φ)cos(φ)P ′ +
+ sin(φ)cos(φ)P′
P + λ sin2 (φ) = m 2
λ sin 2 (φ) − m 2
P = 0
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = m 2
λ sin 2 (φ) − m 2
P = 0
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
sin2
P
(φ)
= 0
sin 2 (φ)P ′′ + sin(φ)cos(φ)P ′ +
logo1
Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = m 2
sin 2 (φ)P ′′ + sin(φ)cos(φ)P ′
+ λ sin 2 (φ) − m 2
P = 0
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
sin2
P
(φ)
= 0
This equation is complicated, because it involves trigonometric
functions.
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Bernd Schröder 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
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin 2 (φ) P′′
+ sin(φ)cos(φ)P′
P P + λ sin2 (φ) = m 2
sin 2 (φ)P ′′ + sin(φ)cos(φ)P ′
+ λ sin 2 (φ) − m 2
P = 0
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
sin2
P = 0
(φ)
This equation is complicated, because it involves trigonometric
functions.
It turns out that the substitution z = cos(φ) will simplify the
equation.
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Bernd Schröder 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
Derivatives for the Substitution
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Bernd Schröder 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
Derivatives for the Substitution
d
dφ P
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Bernd Schröder 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
Derivatives for the Substitution
d
P =
dφ
d
dz P
d
dφ z
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Bernd Schröder 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
Derivatives for the Substitution
d
P =
dφ
d
dz P
d
dφ z
=
d
dz P
d
dφ cos(φ)
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Bernd Schröder 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
Derivatives for the Substitution
d
P =
dφ
=
d
dz P
d
dφ z
d
=
dz P
d
dφ cos(φ)
d
dz P
− sin(φ)
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Bernd Schröder 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
Derivatives for the Substitution
d
P =
dφ
d2 P
dφ 2
=
d
dz P
d
dφ z
d
=
dz P
d
dφ cos(φ)
d
dz P
− sin(φ)
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Bernd Schröder 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
Derivatives for the Substitution
d d
P =
dφ dz P
d
dφ z
d
=
dz P
d
dφ cos(φ)
d
=
dz P
− sin(φ)
d2
d d
P =
dφ 2 dφ dz P
− sin(φ)
logo1
Bernd Schröder 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
Derivatives for the Substitution
d d
P =
dφ dz P
d
dφ z
d
=
dz P
d
dφ cos(φ)
d
=
dz P
− sin(φ)
d2
d d
P =
dφ 2 dφ dz P
− sin(φ)
= d
d
dφ dz P
d
− sin(φ) +
dz P
− cos(φ)
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Bernd Schröder 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
Derivatives for the Substitution
d d
P =
dφ dz P
d
dφ z
d
=
dz P
d
dφ cos(φ)
d
=
dz P
− sin(φ)
d2
d d
P =
dφ 2 dφ dz P
− sin(φ)
= d
d
dφ dz P
d
− sin(φ) +
dz P
− cos(φ)
d d
=
dz dz P
− sin(φ)
d
− sin(φ) +
dz P
− cos(φ)
logo1
Bernd Schröder 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
Derivatives for the Substitution
d d
P =
dφ dz P
d
dφ z
d
=
dz P
d
dφ cos(φ)
d
=
dz P
− sin(φ)
d2
d d
P =
dφ 2 dφ dz P
− sin(φ)
= d
d
dφ dz P
d
− sin(φ) +
dz P
− cos(φ)
d d
=
dz dz P
− sin(φ)
− sin(φ) +
= sin 2 (φ) d2
P − cos(φ)
dz2
d
dz P
d
dz P
− cos(φ)
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Bernd Schröder 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
Generalized Legendre Equation
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Bernd Schröder 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
Generalized Legendre Equation
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
sin2
P = 0
(φ)
logo1
Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2P −cos(φ)dP
dz2 dz
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
sin2
P = 0
(φ)
logo1
Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
dz
2 −cos(φ)dP
+ cos(φ)
sin(φ)
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
dP
− sin(φ)
dz
sin 2 (φ)
P = 0
logo1
Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
dz
2 −cos(φ)dP
+ cos(φ)
sin(φ)
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
dP
− sin(φ) +
dz
sin 2 (φ)
P = 0
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Bernd Schröder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
λ − m2
sin2
P = 0
(φ)

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
dz
2 −cos(φ)dP
+ cos(φ)
sin(φ)
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
dP
− sin(φ) +
dz
sin 2 (φ) d2P − 2cos(φ)dP
dz2 dz +
P = 0
sin2 (φ)
λ − m2
sin2
P = 0
(φ)
λ − m2
sin2
P = 0
(φ)
logo1
Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
2 −cos(φ)dP
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
P = 0
sin2 (φ)
cos(φ) dP
+ − sin(φ) + λ −
dz sin(φ) dz
m2
sin2
P = 0
(φ)
sin 2 (φ) d2P − 2cos(φ)dP
dz2 dz +
λ − m2
sin2
P = 0
(φ)
1 − z 2 d2P dz2 logo1
Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
2 −cos(φ)dP
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
P = 0
sin2 (φ)
cos(φ) dP
+ − sin(φ) + λ −
dz sin(φ) dz
m2
sin2
P = 0
(φ)
sin 2 (φ) d2P − 2cos(φ)dP
dz2 dz +
λ − m2
sin2
P = 0
(φ)
1 − z 2 d2P − 2zdP
dz2 dz
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Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
2 −cos(φ)dP
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
P = 0
sin2 (φ)
cos(φ) dP
+ − sin(φ) + λ −
dz sin(φ) dz
m2
sin2
P = 0
(φ)
sin 2 (φ) d2P − 2cos(φ)dP
dz2 dz +
λ − m2
sin2
P = 0
(φ)
1 − z 2 d2P − 2zdP
dz2 dz +
λ − m2
1 − z2
P
logo1
Bernd Schröder 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
Generalized Legendre Equation
sin 2 (φ) d2 P
dz
2 −cos(φ)dP
P ′′ + cos(φ)
sin(φ) P′
+ λ − m2
P = 0
sin2 (φ)
cos(φ) dP
+ − sin(φ) + λ −
dz sin(φ) dz
m2
sin2
P = 0
(φ)
sin 2 (φ) d2P − 2cos(φ)dP
dz2 dz +
λ − m2
sin2
P = 0
(φ)
1 − z 2 d2P − 2zdP
dz2 dz +
λ − m2
1 − z2
P = 0
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Bernd Schröder 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
Legendre Equations
Let λ be a real number and let m be a nonnegative integer. The
differential equation
1 − x 2
y ′′ − 2xy ′
+ λ − m2
1 − x2
y = 0
is called the generalized Legendre equation.
logo1
Bernd Schröder 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
Legendre Equations
Let λ be a real number and let m be a nonnegative integer. The
differential equation
1 − x 2
y ′′ − 2xy ′
+ λ − m2
1 − x2
y = 0
is called the generalized Legendre equation.
For nonnegative integers l, the differential equation
1 − x 2
y ′′ − 2xy ′ + l(l + 1)y = 0
is called the Legendre equation.
logo1
Bernd Schröder 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
Legendre Equations
Let λ be a real number and let m be a nonnegative integer. The
differential equation
1 − x 2
y ′′ − 2xy ′
+ λ − m2
1 − x2
y = 0
is called the generalized Legendre equation.
For nonnegative integers l, the differential equation
1 − x 2
y ′′ − 2xy ′ + 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.
logo1
Bernd Schröder 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
Legendre Equations
Let λ be a real number and let m be a nonnegative integer. The
differential equation
1 − x 2
y ′′ − 2xy ′
+ λ − m2
1 − x2
y = 0
is called the generalized Legendre equation.
For nonnegative integers l, the differential equation
1 − x 2
y ′′ − 2xy ′ + 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 Schröder 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
Legendre Equations
logo1
Bernd Schröder 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
Legendre Equations
In the Legendre equation
1 − x 2
y ′′ − 2xy ′ + 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:
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Bernd Schröder 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
Legendre Equations
In the Legendre equation
1 − x 2
y ′′ − 2xy ′ + 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.
logo1
Bernd Schröder 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
Legendre Equations
In the Legendre equation
1 − x 2
y ′′ − 2xy ′ + 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 Schröder 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
Legendre Equations
In the Legendre equation
1 − x 2
y ′′ − 2xy ′ + 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 = l(l + 1), the
function u would be infinite on the z-axis, which is not
sensible.
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Bernd Schröder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations