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DLMF: Β§33.14 Definitions and Basic Properties β£ Variables π,Ο΅ β£ Chapter 33 Coulomb Functions
Β§33.14 Definitions and Basic Properties
Contents
Β§33.14(i) Coulomb Wave Equation
Β§33.14(ii) Regular Solution f β‘ ( Ο΅ , β ; r )
Β§33.14(iii) Irregular Solution h β‘ ( Ο΅ , β ; r )
Β§33.14(iv) Solutions s β‘ ( Ο΅ , β ; r ) and
c β‘ ( Ο΅ , β ; r )
Β§33.14(v) Wronskians
Β§33.14(i) Coulomb Wave Equation
Another parametrization of (33.2.1 ) is given by
33.14.1
d 2 w d r 2 + ( Ο΅ + 2 r β β β’ ( β + 1 ) r 2 ) β’ w = 0 ,
where
33.14.2
r
= β Ξ· β’ Ο ,
Ο΅
= 1 / Ξ· 2 .
Again, there is a regular singularity at r = 0 with indices β + 1 and
β β , and an irregular singularity of rank 1 at r = β .
When Ο΅ > 0 the outer turning point is given by
33.14.3
r tp β‘ ( Ο΅ , β ) = ( 1 + Ο΅ β’ β β’ ( β + 1 ) β 1 ) / Ο΅ ;
compare (33.2.2 ).
Β§33.14(ii) Regular Solution f β‘ ( Ο΅ , β ; r )
The function f β‘ ( Ο΅ , β ; r ) is recessive
(Β§2.7(iii) ) at r = 0 , and is defined by
33.14.4
f β‘ ( Ο΅ , β ; r ) = ΞΊ β + 1 β’ M ΞΊ , β + 1 2 β‘ ( 2 β’ r / ΞΊ ) / ( 2 β’ β + 1 ) ! ,
or equivalently
33.14.5
f β‘ ( Ο΅ , β ; r ) = ( 2 β’ r ) β + 1 β’ e β r / ΞΊ β’ M β‘ ( β + 1 β ΞΊ , 2 β’ β + 2 , 2 β’ r / ΞΊ ) / ( 2 β’ β + 1 ) ! ,
where M ΞΊ , ΞΌ β‘ ( z ) and M β‘ ( a , b , z ) are defined in
§§13.14(i) and 13.2(i) , and
33.14.6
ΞΊ = { ( β Ο΅ ) β 1 / 2 , Ο΅ < 0 , r > 0 , β ( β Ο΅ ) β 1 / 2 , Ο΅ < 0 , r < 0 , Β± i β’ Ο΅ β 1 / 2 , Ο΅ > 0 .
The choice of sign in the last line of (33.14.6 ) is immaterial:
the same function f β‘ ( Ο΅ , β ; r ) is obtained. This is a
consequence of Kummerβs transformation (Β§13.2(vii) ).
f β‘ ( Ο΅ , β ; r ) is real and an analytic function of r in the interval
β β < r < β , and it is also an analytic function of Ο΅ when
β β < Ο΅ < β . This includes Ο΅ = 0 , hence
f β‘ ( Ο΅ , β ; r ) can be expanded in a convergent power series in
Ο΅ in a neighborhood of Ο΅ = 0 (Β§33.20(ii) ).
Β§33.14(iii) Irregular Solution h β‘ ( Ο΅ , β ; r )
For nonzero values of Ο΅ and r
the function h β‘ ( Ο΅ , β ; r ) is defined by
33.14.7
h β‘ ( Ο΅ , β ; r ) = Ξ β‘ ( β + 1 β ΞΊ ) Ο β’ ΞΊ β β’ ( W ΞΊ , β + 1 2 β‘ ( 2 β’ r / ΞΊ ) + ( β 1 ) β β’ S β‘ ( Ο΅ , r ) β’ Ξ β‘ ( β + 1 + ΞΊ ) 2 β’ ( 2 β’ β + 1 ) ! β’ M ΞΊ , β + 1 2 β‘ ( 2 β’ r / ΞΊ ) ) ,
where ΞΊ is given by (33.14.6 ) and
33.14.8
S β‘ ( Ο΅ , r ) = { 2 β’ cos β‘ ( Ο β’ | Ο΅ | β 1 / 2 ) , Ο΅ < 0 , r > 0 , 0 , Ο΅ < 0 , r < 0 , e Ο β’ Ο΅ β 1 / 2 , Ο΅ > 0 , r > 0 , e β Ο β’ Ο΅ β 1 / 2 , Ο΅ > 0 , r < 0 .
(Again, the choice of the ambiguous sign in the last line of
(33.14.6 ) is immaterial.)
h β‘ ( Ο΅ , β ; r ) is real and an analytic function
of each of r and Ο΅ in the intervals
β β < r < β and β β < Ο΅ < β ,
except when r = 0 or Ο΅ = 0 .
Β§33.14(iv) Solutions s β‘ ( Ο΅ , β ; r ) and
c β‘ ( Ο΅ , β ; r )
The functions s β‘ ( Ο΅ , β ; r ) and
c β‘ ( Ο΅ , β ; r ) are defined by
33.14.9
s β‘ ( Ο΅ , β ; r )
= ( B β‘ ( Ο΅ , β ) / 2 ) 1 / 2 β’ f β‘ ( Ο΅ , β ; r ) ,
c β‘ ( Ο΅ , β ; r )
= ( 2 β’ B β‘ ( Ο΅ , β ) ) β 1 / 2 β’ h β‘ ( Ο΅ , β ; r ) ,
where
33.14.10
B β‘ ( Ο΅ , β ) = { A β‘ ( Ο΅ , β ) β’ ( 1 β exp β‘ ( β 2 β’ Ο / Ο΅ 1 / 2 ) ) β 1 , Ο΅ > 0 , A β‘ ( Ο΅ , β ) , Ο΅ β€ 0 ,
and
33.14.11
A β‘ ( Ο΅ , β ) = β k = 0 β ( 1 + Ο΅ β’ k 2 ) .
An alternative formula for A β‘ ( Ο΅ , β ) is
33.14.12
A β‘ ( Ο΅ , β ) = Ξ β‘ ( 1 + β + ΞΊ ) Ξ β‘ ( ΞΊ β β ) β’ ΞΊ β 2 β’ β β 1 ,
the choice of sign in the last line of (33.14.6 ) again being
immaterial.
When Ο΅ < 0 and β > ( β Ο΅ ) β 1 / 2 the quantity
A β‘ ( Ο΅ , β ) may
be negative, causing s β‘ ( Ο΅ , β ; r ) and
c β‘ ( Ο΅ , β ; r ) to become imaginary.
The function s β‘ ( Ο΅ , β ; r ) has the following properties:
33.14.13
β« 0 β s β‘ ( Ο΅ 1 , β ; r ) β’ s β‘ ( Ο΅ 2 , β ; r ) β’ d r = Ξ΄ β‘ ( Ο΅ 1 β Ο΅ 2 ) ,
Ο΅ 1 , Ο΅ 2 > 0 ,
where the right-hand side is the Dirac delta (Β§1.17 ). When
Ο΅ = β 1 / n 2 , n = β + 1 , β + 2 , β¦ ,
s β‘ ( Ο΅ , β ; r ) is exp β‘ ( β r / n ) times a polynomial in r / n , and
33.14.14
Ο n , β β‘ ( r ) = ( β 1 ) β + 1 + n β’ ( 2 / n 3 ) 1 / 2 β’ s β‘ ( β 1 / n 2 , β ; r ) = ( β 1 ) β + 1 + n n β + 2 β’ ( ( n β β β 1 ) ! ( n + β ) ! ) 1 / 2 β’ ( 2 β’ r ) β + 1 β’ e β r / n β’ L n β β β 1 ( 2 β’ β + 1 ) β‘ ( 2 β’ r / n )
satisfies
33.14.15
β« 0 β Ο m , β β‘ ( r ) β’ Ο n , β β‘ ( r ) β’ d r = Ξ΄ m , n .
Note that the functions Ο n , β , n = β , β + 1 , β¦ , do not form a complete orthonormal system.
Β§33.14(v) Wronskians
With arguments Ο΅ , β , r suppressed,
33.14.16
π² β‘ { h , f }
= 2 / Ο ,
π² β‘ { c , s }
= 1 / Ο .