separation of variables -- legendre equations...solution technique for partial differential...
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![Page 1: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/1.jpg)
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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables – LegendreEquations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 2: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/2.jpg)
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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables1. Solution technique for partial differential equations.
2. If the unknown function u depends on variables ρ,θ ,φ , weassume there is a solution of the form u = R(ρ)T(θ)P(φ).
3. The special form of this solution function allows us toreplace the original partial differential equation withseveral 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 usefulresults for the partial differential equations.
6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 3: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/3.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables1. 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 toreplace the original partial differential equation withseveral 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 usefulresults for the partial differential equations.
6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 4: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/4.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables1. 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 withseveral 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 usefulresults for the partial differential equations.
6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 5: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/5.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables1. 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 withseveral 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 obtainmust typically be processed some more to give usefulresults for the partial differential equations.
6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 6: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/6.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables1. 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 withseveral 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 usefulresults for the partial differential equations.
6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 7: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/7.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separation of Variables1. 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 withseveral 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 usefulresults for the partial differential equations.
6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 8: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/8.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
How Deep?
plus about 200 pages of reallyawesome functional analysis.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 9: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/9.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
How Deep?
plus about 200 pages of reallyawesome functional analysis.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 10: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/10.jpg)
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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 theLaplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.
2. The time independent Schrodinger equation
− h2m
∆φ +Vφ = Eφ describes certain quantummechanical 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 potentialand E is the energy eigenvalue.
3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 11: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/11.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
The Equation ∆u = f (ρ)u1. For constant f , this is an eigenvalue equation for the
Laplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.
2. The time independent Schrodinger equation
− h2m
∆φ +Vφ = Eφ describes certain quantummechanical 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 potentialand E is the energy eigenvalue.
3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 12: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/12.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
The Equation ∆u = f (ρ)u1. For constant f , this is an eigenvalue equation for the
Laplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.
2. The time independent Schrodinger equation
− h2m
∆φ +Vφ = Eφ describes certain quantummechanical 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 potentialand E is the energy eigenvalue.
3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 13: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/13.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
The Equation ∆u = f (ρ)u1. For constant f , this is an eigenvalue equation for the
Laplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.
2. The time independent Schrodinger equation
− h2m
∆φ +Vφ = Eφ describes certain quantummechanical 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 potentialand E is the energy eigenvalue.
3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 14: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/14.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)
∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 15: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/15.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 16: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/16.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP
+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 17: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/17.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP
+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 18: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/18.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′
+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 19: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/19.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′
+1
ρ2 sin2(φ)RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 20: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/20.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P
= f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 21: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/21.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 22: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/22.jpg)
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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R
+2ρR′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 23: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/23.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R
+P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 24: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/24.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P
+cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 25: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/25.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P
+1
sin2(φ)T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 26: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/26.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T
= ρ2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 27: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/27.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 28: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/28.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 29: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/29.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 30: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/30.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 31: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/31.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 32: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/32.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0.
(QM: Laguerre polys.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 33: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/33.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +
2ρ
∂u∂ρ
+1
ρ2∂ 2u∂φ 2 +
cos(φ)ρ2 sin(φ)
∂u∂φ
+1
ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u
R′′TP+2ρ
R′TP+1
ρ2 RTP′′+cos(φ)
ρ2 sin(φ)RTP′+
1ρ2 sin2(φ)
RT ′′P = f (ρ)RTP
ρ2 R′′
R+2ρ
R′
R+
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)
Bring all terms that depend on ρ to the right side:
P′′
P+
cos(φ)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= ρ
2f (ρ)−ρ2 R′′
R−2ρ
R′
R,
Both sides must be constant.
ρ2f (ρ)−ρ
2 R′′
R−2ρ
R′
R=−λ , or
ρ2R′′+2ρR′−
(λR+ρ
2f (ρ))
R = 0. (QM: Laguerre polys.)Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 34: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/34.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 35: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/35.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 36: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/36.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 37: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/37.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
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 anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 38: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/38.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 39: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/39.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 40: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/40.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic.
Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 41: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/41.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.
So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 42: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/42.jpg)
logo1
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)sin(φ)
P′
P+
1sin2(φ)
T ′′
T= −λ
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+
T ′′
T= −λ sin2(φ)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = −T ′′
TBoth sides must be constant.
−T ′′
T= c leads to T ′′+ cT = 0.
But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 43: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/43.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = m2
sin2(φ)P′′+ sin(φ)cos(φ)P′+(
λ sin2(φ)−m2)
P = 0
P′′+cos(φ)sin(φ)
P′+(
λ − m2
sin2(φ)
)P = 0
This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 44: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/44.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = m2
sin2(φ)P′′+ sin(φ)cos(φ)P′+(
λ sin2(φ)−m2)
P = 0
P′′+cos(φ)sin(φ)
P′+(
λ − m2
sin2(φ)
)P = 0
This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 45: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/45.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = m2
sin2(φ)P′′+ sin(φ)cos(φ)P′+(
λ sin2(φ)−m2)
P = 0
P′′+cos(φ)sin(φ)
P′+(
λ − m2
sin2(φ)
)P = 0
This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 46: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/46.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = m2
sin2(φ)P′′+ sin(φ)cos(φ)P′+(
λ sin2(φ)−m2)
P = 0
P′′+cos(φ)sin(φ)
P′+(
λ − m2
sin2(φ)
)P = 0
This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 47: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/47.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = m2
sin2(φ)P′′+ sin(φ)cos(φ)P′+(
λ sin2(φ)−m2)
P = 0
P′′+cos(φ)sin(φ)
P′+(
λ − m2
sin2(φ)
)P = 0
This equation is complicated, because it involves trigonometricfunctions.
It turns out that the substitution z = cos(φ) will simplify theequation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 48: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/48.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Separating the Equation ∆u = f (ρ)u (Polar Part)
sin2(φ)P′′
P+ sin(φ)cos(φ)
P′
P+λ sin2(φ) = m2
sin2(φ)P′′+ sin(φ)cos(φ)P′+(
λ sin2(φ)−m2)
P = 0
P′′+cos(φ)sin(φ)
P′+(
λ − m2
sin2(φ)
)P = 0
This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 49: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/49.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitution
ddφ
P =(
ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 50: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/50.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP
=(
ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 51: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/51.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 52: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/52.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 53: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/53.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 54: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/54.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P
=d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 55: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/55.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 56: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/56.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 57: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/57.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 58: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/58.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Derivatives for the Substitutiond
dφP =
(ddz
P)(
ddφ
z)
=(
ddz
P)(
ddφ
cos(φ))
=(
ddz
P)(
− sin(φ))
d2
dφ 2 P =d
dφ
((ddz
P)(
− sin(φ)))
=d
dφ
(ddz
P)(
− sin(φ))+
(ddz
P)(
− cos(φ))
=[
ddz
(ddz
P)(
− sin(φ))](
− sin(φ))+
(ddz
P)(
− cos(φ))
= sin2(φ)d2
dz2 P− cos(φ)(
ddz
P)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 59: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/59.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 60: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/60.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 61: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/61.jpg)
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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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 62: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/62.jpg)
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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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)
+(
λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 63: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/63.jpg)
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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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 64: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/64.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0
(1− z2
) d2Pdz2 −2z
dPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 65: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/65.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2
−2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 66: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/66.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 67: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/67.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P
= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 68: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/68.jpg)
logo1
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(φ)
)P = 0
sin2(φ)d2Pdz2 −cos(φ)
dPdz
+cos(φ)sin(φ)
dPdz
(− sin(φ)
)+
(λ− m2
sin2(φ)
)P = 0
sin2(φ)d2Pdz2 −2cos(φ)
dPdz
+(
λ − m2
sin2(φ)
)P = 0(
1− z2) d2P
dz2 −2zdPdz
+(
λ − m2
1− z2
)P = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 69: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/69.jpg)
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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(
1− x2)
y′′−2xy′+(
λ − m2
1− x2
)y = 0
is called the generalized Legendre equation.
For nonnegative integers l, the differential equation(1− x2
)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 theequation for T(θ) in the separation of variables.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 70: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/70.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(
1− x2)
y′′−2xy′+(
λ − m2
1− x2
)y = 0
is called the generalized Legendre equation.For nonnegative integers l, the differential equation(
1− x2)
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 theequation for T(θ) in the separation of variables.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 71: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/71.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(
1− x2)
y′′−2xy′+(
λ − m2
1− x2
)y = 0
is called the generalized Legendre equation.For nonnegative integers l, the differential equation(
1− x2)
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 theequation for T(θ) in the separation of variables.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 72: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/72.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(
1− x2)
y′′−2xy′+(
λ − m2
1− x2
)y = 0
is called the generalized Legendre equation.For nonnegative integers l, the differential equation(
1− x2)
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 theequation for T(θ) in the separation of variables.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 73: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/73.jpg)
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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre Equations
In the Legendre equation(1− x2
)y′′−2xy′+ l(l+1)y = 0,
the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:
1. For λ not of this form the solutions go to infinity as zapproaches ±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), thefunction u would be infinite on the z-axis, which is notsensible.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 74: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/74.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsIn the Legendre equation(
1− x2)
y′′−2xy′+ l(l+1)y = 0,
the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:
1. For λ not of this form the solutions go to infinity as zapproaches ±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), thefunction u would be infinite on the z-axis, which is notsensible.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 75: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/75.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsIn the Legendre equation(
1− x2)
y′′−2xy′+ l(l+1)y = 0,
the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:
1. For λ not of this form the solutions go to infinity as zapproaches ±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), thefunction u would be infinite on the z-axis, which is notsensible.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 76: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/76.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsIn the Legendre equation(
1− x2)
y′′−2xy′+ l(l+1)y = 0,
the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:
1. For λ not of this form the solutions go to infinity as zapproaches ±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), thefunction u would be infinite on the z-axis, which is notsensible.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations
![Page 77: Separation of Variables -- Legendre Equations...Solution technique for partial differential equations. 2. If the unknown function u depends on variables ρ,θ,φ, we assume there is](https://reader035.vdocuments.site/reader035/viewer/2022062417/61478f74afbe1968d37a212c/html5/thumbnails/77.jpg)
logo1
Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation
Legendre EquationsIn the Legendre equation(
1− x2)
y′′−2xy′+ l(l+1)y = 0,
the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:
1. For λ not of this form the solutions go to infinity as zapproaches ±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), thefunction u would be infinite on the z-axis, which is notsensible.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Separation of Variables – Legendre Equations