SPECIAL FUNCTIONS

SPECIAL FUNCTIONS

I n this appendix we list a number of properties of various special functions that have proved useful in this text.

Legendre polynomials Ρη(μ) are defined (References 1-6) by the following power series expansion:

W 2" ^{1 }} k\(n

- *)! (n -

Legendre polynomials are generated by the following power series expansion

(1 - 2 V + A²) -¹'² = % Η«Ρ„(μ), (D.2) 71=0

where h < 1.

T h e Legendre polynomials satisfy the following so-called addition theorem:

where μ0 , μ, μ/, φ and are related to each other by

μ0 = μμ' + V l - μ² V l - j * '² COS(<p - φ') , (D.4) and Ρβη(μ) is the associated Legendre polynomial of order β and degree n.

T h e quantities /x0, /x, and μ/ may be interpreted as the cosines of certain angles related to each other and to the angles ψ and φ' as shown in Fig. D . l . T h e relation ( D . 3 ) is then the familiar one connecting the angles and sides of a spherical triangle. T h e associated Legendre poly­

nomials are defined by

Pfr) = 0 - M²) "^{/ 2} [ J » ] · ( D. 5 ) 325

( D . 6 )

T h e associated Legendre polynomials satisfy the following orthogonality relation

J > ^P » » = ( 2 , + 1) t^P - W ^{8 nN} ( D . 7 )

and are connected to each other by various recurrence relations of which the following are useful in the text:

(η - β + 1) Ρβη+1(μ) - (In + \)μ Ρ*(/χ) + (η + β) Ρβη_,(μ) = 0 , ( D . 8 )

( 1 - μ ² ) ^ Ρ » = ( » + ! ) / * Ρ^βΜ - ( η - β + l ) P f » , ( D . 9 )

= (η + β)Ρβ_1(μ)-ημΡ?η(μ). ( D . 1 0 )

F I G . D . l . Geometrical definitions of θ, θ\ 0O, φ, and φ'.

T h e relation

[ 0 ,

2^ln\[(n +

+ / + ! ) ! [ ( » - / ) / 2 ] ! '

if η — I < 0 or odd , if η — I ^ 0 and even ,

( D. l l )

is sometimes useful. M a n y more properties of these functions are known (see References 1-6).

A special case is that for which β = 0:

dz ^Lk(z). (D.15) Other normalizations are frequently used. M a n y other properties of

these functions are known.

Special polynomials of use in the application of the Spencer-Fano method are as follows (see Reference 7 ) :

u>W-2ijl idz J £0k\(j-k)\2^k ' ^{{ , 1 0;}

T h e first few polynomials are:

U0 = 1 ,

U2(z)=Uz*-5z + 3),

TO = ^ (z> - 12»· + 33* - 15),

TO =4"'

TO=^(*-3),

TO=^(*²-9* + 15),

Va{z) =

m

~

*

* ~

·

(D.18) Laguerre polynomials are occasionally useful (References 4-6). T h e y are defined by the relation

( D-^{1 2)}

and satisfy the orthogonality relation

fdxer'LAz)L1ILx)=8it (D.13)

J 0

and the recurrence relation

U + l)LM{*) - (2j + 1 - z)Llz) +JLU*) = 0 . (D.14) Also,

dL^z) _

T h e y satisfy the following integral relations:

Γ dz z^2k e-W Ε/χ| ζ

I)

Γ dz z^+v e-M Vl\ ζ I) = 0, if k <j (D.19)

J -00

and the differential equations

ζ

d³Uj , /t , Nd f 2t / , , 1X dU, „.rr

^ ⁺ ^ ~ ^{3 z}^ ⁺²^ - ^ - d f - ²^ = ⁰^

dW d²V dV

' 0

and the relations

fd*e-Ujiz)U#z) = Sik ,

J 0

Γ ώ Γ Ψ/ , ) Γ ί( « ) = ίΛ ,

^ η

(D.20)

T h e polynomials Uf and F}t

i W & ( 2 * ) ! *l ( / - * ) ' '

and (D.21)

, W a( 2* - l) t * ! 0* + l - * ) > * satisfy the relations

(D.22)

(D.23)

References

1. Bateman, H., "Partial Differential Equations of Mathematical Physics." Dover, N e w York, 1944, Chapter 6.

2. Whittaker, Ε. T., and Watson, G . N., " A Course of Modern Analysis."

Cambridge Univ. Press, London and New York, 1940, Chapter 15.

3. Copson, Ε. T., " A n Introduction to the Theory of Functions of a Complex Variable.'' Oxford Univ. Press (Clarendon), London and N e w York, 1935, Chapter 11.

4. Morse, P. M . , and Feshbach, H., "Methods of Theoretical Physics." M c G r a w - Hill, N e w York, 1953, Chapters 5, 6, 10.

5. Courant, R., and Hubert, D., "Methoden der mathematischen Physik,"

2nd edition, Vol. I, Chapters 2, 5; Vol. II, Chapter 4. Springer, Berlin, 1931.

6. Frank, P., and von Mises, R. " D i e Differential und Integralgleichungen der Mechanik undPhysik," 2nd edition, Vol. I, Chapter 8. Vieweg, Braunsch­

weig, Germany, 1930.

7. Spencer, L. V. and Fano, U., J. Research Natl. Bur. Standards 46, 446 (1951).