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

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