Lecture 1 phys 404 Legendre Functions · Legendre Functions – physical justification • Eq. 3...
Transcript of Lecture 1 phys 404 Legendre Functions · Legendre Functions – physical justification • Eq. 3...
PHYS 404 Lecture 1: Legendre Functions
Dr. Vasileios Lempesis
PHYS404-LECTURE1DR.V.LEMPESIS 1
Legendre Functions – physical justification
• Legendre functions or Legendre polynomials are the solutions of Legendre’s differential equation that appear when we separate the variables of HelmholG’ equation, Laplace equation or Schrodinger equation using spherical coordinates.
PHYS404-LECTURE1DR.V.LEMPESIS 2
Legendre Functions – physical justification
• Almost all the cases are specific cases of the general differential equation:
• Following the method of separation of variables, i.e. by considering a solution of the form:
∇2 + k 2 −U (r){ }Ψ r( ) = 0, k = const (1)
Ψ r( ) = R r( )Y θ ,ϕ( ) (2)
PHYS404-LECTURE1DR.V.LEMPESIS 3
Legendre Functions – physical justification
• The differential equation (1) splits in two simpler equations
1sinθ
∂∂θ
sinθ ∂∂θ
"
#$
%
&'+
1sin2θ
∂2
∂φ 2
()*
+,-Y θ ,ϕ( ) = −λY θ ,ϕ( ) (1.3)
1r2
ddrr2 dRdr
⎛
⎝⎜
⎞
⎠⎟+ k 2 −U (r)− λ
r2
⎡
⎣⎢
⎤
⎦⎥R = 0 (1.4)
PHYS404-LECTURE1DR.V.LEMPESIS 4
Legendre Functions – physical justification
• Eq. 3 is independent from function U(r) and is called angular equation. Solutions of this equation could be: trigonometric functions, Legendre functions, associated Legendre functions and the spherical harmonics.
• Eq. 4 does depend on function U(r) and is called radial equation. Solutions of this equation could be: Bessel functions, Hermite functions and Laguerre functions.
PHYS404-LECTURE1DR.V.LEMPESIS 5
The angular equation • Try to consider for Eq. 3 a solution of the
separating variable form:
• Then the radial equation is split into two equations:
Y θ ,ϕ( ) =Θ θ( )Φ ϕ( ) (1.5)
1sinθ
ddθ
sinθ dΘdθ
"
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%
&'+ λ −
µ 2
sin2θ
"
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%
&'Θ = 0 (1.6)
d 2Φ
d 2φ+µ 2Φ = 0 (1.7)
PHYS404-LECTURE1DR.V.LEMPESIS 6
The associated Legendre equation
• If we make the substitutions
Then Eq. 1.7 takes the formThis is the so called associated Legendre equation
x = cosθ , u(x) =Θ θ( ), λ=ν (ν +1) (1.8)
ddx
1− x2( ) dudx"
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%
&'+ ν (ν +1)− µ 2
1− x2
"
#$
%
&'u = 0 (1.9)
Substitution x=cosθ is needed in order to give us a DE with rational coefficients for which we have “efficient tools” for its solution; method of power series PHYS404-LECTURE1DR.V.LEMPESIS 7
The associated Legendre equation
• The associated Legendre Equation has two linearly independent solutions:
• The associated Legendre function of first kind:• The associated Legendre function of second
kind: • In physical problems Eq. 1.7 has single valued
and periodical solutions only if m =integer. In this case
Pµν x( )
Qµν x( )
Φm = exp imφ( ) / 2π (1.10)
PHYS404-LECTURE1DR.V.LEMPESIS 8
The associated Legendre equation
• Then in this case the associated Legendre Equation is wriZen as
ddx
1− x2( ) dudx"
#$
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&'+ ν (ν +1)− m2
1− x2
"
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&'u = 0 (1.10)
PHYS404-LECTURE1DR.V.LEMPESIS 9
Legendre Diff. Equation• If in the previous equation we consider m=0
and also consider that then we get the Legendre differential equation:
• Eq. 1.11 has finite solutions at the points where only if is a positive integer.
u(x)m=0
= y(x)
ddx
1− x2( ) dydx"
#$
%
&'+ν (ν +1)y = 0 (1.11)
x = ±1 ν = n
PHYS404-LECTURE1DR.V.LEMPESIS 10
Legendre Functions• Legendre functions (or polynomials) ,
are a solution of Legendre differential equation about the origin (x = 0).
• The polynomials are rarely used in physics problems so we are not going to deal with them further.
• On the contrary the polynomials , which appear in many physical problems, may be defined by the so called generating function.
Pn x( ) Qn x( )
Pℓ x( )
Qℓ x( )
PHYS404-LECTURE1DR.V.LEMPESIS 11
Legendre Polynomial The generating function
The generating function of Legendre polynomials: (1.12) Which has an important application in electric multipole
expansions. If we expand this function as a binomial series if we obtain
(1.13)t <1
PHYS404-LECTURE1DR.V.LEMPESIS 12
Legendre Polynomial The generating function
Using the binomial theorem we can expand the generating function as follows:
From the above series we can get the values of the
Legendre functions.
g(t,x) = 1− 2xt + t2( )−1/2
=2n( )!
22n n!( )2
2xt − t2( )n
n=0
∞
∑ (1.14)
PHYS404-LECTURE1DR.V.LEMPESIS 13
Legendre Polynomial plots
PHYS404-LECTURE1DR.V.LEMPESIS 14
The Recurrence Relations
Legendre polynomials obey the following recurrence relations:
(1.15) These relations are valid for
2n+1( ) xPn (x) = n+1( )Pn+1(x)+ nPn−1(x)Pn+1' (x)+ Pn−1
' (x) = 2xPn' (x) + Pn (x)
Pn+1' (x)− Pn−1
' (x) = 2n+1( )Pn (x)Pn+1' (x) = n+1( )Pn (x)+ xPn' (x)Pn−1' (x) = −nPn (x)+ xPn
' (x)
1− x2( )Pn' (x) = nPn−1(x)− nxPn (x)PHYS404-LECTURE1DR.V.LEMPESIS 15
Legendre Polynomials Special Properties
• Some special values • (1.6a) (1.16b) and Where (called double factorial)
PHYS404-LECTURE1DR.V.LEMPESIS 16
Legendre Polynomials Special Properties
• The Parity property:(with respect to ) (1.17a) (1.17b) If n is odd the parity of the polynomial is odd, but if it is
even the parity of the polynomial is even. • Upper and lower Bounds for (1.18)
PHYS404-LECTURE1DR.V.LEMPESIS 17
Legendre Polynomials Orthogonality
Legendre’s equation is a self-adjoint equation, which satisfies Sturm-Liouiville theory, where the solutions are expected to be orthogonal to satisfying certain boundary conditions. Legendre polynomials are a set of orthogonal functions on (-1,1). (1.19)where
PHYS404-LECTURE1DR.V.LEMPESIS 18
Legendre Polynomials Legendre Series
• According to Sturm-Liouville theory that Legendre polynomial form a complete set.
• (1.20)
• The coefficients are obtained by multiplying the series by and integrating in the interval [-1,1] (1.21)
PHYS404-LECTURE1DR.V.LEMPESIS 19
an =2n+12
f (x)Pn (x)−1
1
∫
Alternate definitions of Legendre polynomials
• Legendre polynomials can be defined with the help of the so called Rodriguez formula:
(1.22)• The Rodrigues’ formula provides a means of
developing a means of developing an integral representation in the complex plane with the Schlaefli integral :
Pn (x) =12n n!
ddx
!
"#
$
%&
n
x2 −1( )n
Pn (z) =12n n!
ddz!
"#
$
%&
n
z2 −1( )n=2−n
2πi
t2 −1( )n
t − z( )n+1!∫ dt Contour
encloses t=z.PHYS404-LECTURE1DR.V.LEMPESIS 20