Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

Скачать файл
Заказать решение

ВУЗ: Не указан

Категория: Не указан

Дисциплина: Не указана

Добавлен: 28.06.2024

Просмотров: 837

Скачиваний: 0

ВНИМАНИЕ! Если данный файл нарушает Ваши авторские права, то обязательно сообщите нам.

A.1 Spherical Bessel Functions

255

 

2n + 1

zn

(x) = zn−1(x) + zn+1(x),

 

 

(A.6)

 

x

 

 

(2n + 1) z

(x) = nz

n−1

(x)

−

(n + 1)z

n+1

(x),

(A.7)

 

 

n

 

 

 

 

 

and

d

dx [xzn(x)] = xzn−1(x) − nzn(x), (A.8)

where zn stands for any spherical function. The Wronskian relation for the spherical Bessel and Neumann functions is

 

jn(x)yn

(x) − jn(x)yn(x) =

 

1

 

 

 

 

 

 

x2

 

 

 

 

and for small values of the argument we have

 

 

 

 

 

 

 

 

 

jn(x) =

 

 

 

xn

1 + O x2 ,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2n + 1)!!

 

 

 

 

 

 

 

 

 

 

 

yn(x) =

 

 

(2n − 1)!!

1 + O x2

 

 

 

 

 

 

 

 

−

xn+1

 

 

 

 

 

 

 

 

 

 

as x → 0, while for large value of the argument,

 

 

 

 

 

 

 

 

jn(x) =

1

cos

 

 

 

 

(n + 1)π

!

 

 

 

 

1

"

 

 

x −

 

 

 

 

 

 

1 + O

 

 

 

,

x

 

2

 

 

 

x

yn(x) =

1

sin

 

 

 

(n + 1)π

!

 

 

 

 

1

 

"

 

 

x −

 

 

 

 

 

1 + O

 

 

,

x

 

2

 

 

x

(1)

(x) =

ej[x−(n+1)π/2] !

1

"

 

 

 

 

hn

 

 

 

 

 

 

 

 

 

 

1 + O

 

 

 

,

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

h(2)

(x) =

e−j[x−(n+1)π/2]

!1 + O

1

"

 

 

 

 

n

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

as x → ∞.

Spherical Bessel functions can be expressed in terms of trigonometric func-

tions, and the first few spherical functions have the explicit forms

 

j0

(x) =

sin x

,

 

 

 

 

 

 

 

(A.9)

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

y0

(x) = −

cos x

 

 

 

 

 

 

(A.10)

 

 

,

 

 

 

 

 

 

x

 

 

 

 

 

 

 

j1

(x) =

sin x

−

cos x

,

 

(A.11)

 

 

 

 

 

 

 

x2

 

x

 

(x) = −

cos x

 

sin x

(A.12)

y1

 

 

−

 

 

 

,

x2

 

 

x

 
jn(x)

256 A Spherical Functions

and

h

(1)

(x) =

 

ejx

,

 

0

 

jx

 

 

 

 

 

 

 

 

 

 

 

 

 

(2)

(x) =

−

e−jx

h0

 

 

.

 

jx

The spherical Bessel functions are computed by downward recursion. This recursion begins with two successive functions of small values and produces functions proportional to the Bessel functions rather than the actual Bessel functions. The constant of proportionality between the two sets of functions is obtained from the function of order zero j0. Alternatively, the spherical Bessel functions can be computed with an algorithm involving the auxiliary function χn [169]

χn(x) = jn−1(x) .

In this case, the functions χn are calculated with the downward recurrence relation

1

χn(x) = 2nx+1 − χn+1(x) ,

the constant of proportionality is obtained from the function of order one, χ1(x) = 1/x − cot x, and the spherical Bessel functions are computed with the upward recursion

jn(x) = χn(x)jn−1(x)

starting at j1. The spherical Neumann functions are calculated by upward recursion starting with the functions of order zero and one, y0 and y1, respectively.

A.2 Legendre Functions

With the substitution x = cos θ, the associated Legendre equation transforms to

 

 

m2

 

 

1 − x2 f (x) − 2xf (x) + n(n + 1) −

 

 

 

f (x) = 0.

(A.13)

1

−

x2

 

 

 

 

 

This equation is characterized by regular singularities at the points x = ±1 and at infinity. For m = 0, there are two linearly independent solutions to the Legendre di erential equation and these solutions can be expressed as power series about the origin x = 0. In general, these series do not converge for

A.2 Legendre Functions

257

x = ±1, but if n is a positive integer, one of the series breaks o after a finite number of terms and has a finite value at the poles. These polynomial solutions are called Legendre polynomials and are denoted by Pn(x). For m = 0, the solutions to (A.13) which are finite at the poles x = ±1 are the associated Legendre functions. If m and n are integers, the associated Legendre functions are defined as

 

 

 

 

 

 

 

 

 

P m(x) = 1

−

x2 m/2

dmPn(x)

.

 

 

 

 

 

 

 

 

 

 

 

n

 

 

 

dxm

 

 

 

Useful recurrence relations satisfied by Pnm are

 

 

 

 

 

 

(2n + 1) xPnm(x) = (n − m + 1) Pnm+1(x) + (n + m) Pnm−1(x),

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(A.14)

 

 

 

 

 

 

 

 

(2n + 1)

 

1 − x2

Pnm(x) = (n + m − 1) (n + m) Pnm−−11(x)

(A.15)

 

 

 

 

 

 

 

 

 

 

 

 

 

− (n − m + 1) (n − m + 2) Pnm+1−1(x),

 

(2n + 1)

1

−

x2P m−1

(x) = P m (x)

 

P m

(cos θ) ,

(A.16)

 

 

 

 

 

 

 

n

 

n+1

− n−1

 

 

 

 

 

(2n + 1) 1

−

x2

d

P m(x) = (n + 1) (n + m) P m

 

(x)

 

 

 

 

 

 

 

 

 

 

 

dx n

 

 

 

 

 

 

n−1

 

 

 

 

 

 

 

 

 

d

 

 

 

−n (n − m + 1) Pnm+1(x),

(A.17)

1

 

−

x2

P m

(x) = (n + m)P m

(x)

−

nxP m(x).

(A.18)

 

 

 

 

 

 

 

dx n

 

 

 

n−1

 

 

 

n

 

The angular functions πm and τ m are related to the associated Legendre

n n functions by the relations

 

 

 

m

 

 

 

 

Pnm(cos θ)

 

 

 

 

(A.19)

 

 

 

πn (θ) =

 

 

sin θ

 

 

,

 

 

 

 

 

 

 

 

τnm(θ) =

d

Pnm(cos θ).

 

 

(A.20)

 

 

 

dθ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For positive values of m and for θ → 0 or θ → π,

 

 

 

 

 

m

(cos θ)

→

(n + m)!

2

 

 

m

 

2n + 1

 

Pn

 

 

 

 

 

 

 

 

 

 

 

Jm

 

 

θ

(n

−

m)!

2n + 1

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m

(θ) →

1

 

 

 

(n + m)!

 

 

 

 

θ

m−1

 

 

mπn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,

 

 

2

(n

−

m)!(m

−

1)!

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m−1

 

 

m

(θ) →

1

 

 

 

(n + m)!

 

 

 

 

θ

 

 

τn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,

 

 

2

 

(n

−

m)!(m

−

1)!

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

258 A Spherical Functions

while for n 1 and away from θ = 0 and θ = π, we have

 

 

 

 

 

 

2

 

1

 

 

 

 

 

 

2n + 1

 

 

 

 

 

 

 

 

 

 

 

 

P m(cos θ)

→

2

(sin θ)− 21

nm− 21 cos

 

θ +

mπ

 

−

 

 

π

 

 

 

π

 

 

 

2

 

 

 

 

 

4

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

1

 

 

nm− 21 cos

2n + 1

 

 

 

 

 

 

 

 

 

 

,

πm(θ)

 

2

(sin θ)− 23

θ +

mπ

 

 

π

→

 

π

 

 

 

 

2

 

2 −

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

 

 

2

 

 

1

 

 

 

 

 

2n + 1

 

 

 

 

 

 

 

 

 

 

 

 

 

m

 

 

2

 

− 21

 

m+ 21

θ +

mπ π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

τn

(θ)

→

 

π

 

 

 

(sin θ)

n

 

sin

 

2

 

2 −

4 .

In our analysis we use the associated Legendre functions with positive values of the index m. For m ≥ 0, the normalized associated Legendre functions are given by

Pnm(cos θ) = cmnPnm(cos θ),

where cmn is a normalization constant and

 

 

 

 

 

 

 

 

 

 

c

 

=

2n + 1

 

(n − m)!

.

 

 

 

 

 

2

 

 

mn

 

· (n + m)!

Similarly, the normalized angular functions πm and τ m

m

m

n

n

 

 

angular functions πn

and τn by the relations

 

 

πnm(θ) = cmnπnm(θ),

 

 

m

m

 

 

τn

(θ) = cmnτn (θ) .

 

(A.21)

are related to the

(A.22)

(A.23)

An algorithm for computing the normalized associated Legendre functions Pnm involves the following steps.

(1) For m = 0, compute Pn+1 by using the recurrence relation

Pn+1(cos θ) =

1

 

 

 

 

 

 

 

 

 

 

 

 

 

cos θPn(cos θ)

(2n + 1)(2n + 3)

 

n + 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n

 

2n + 3

n ≥ 1

 

−

 

 

 

 

 

 

 

 

 

Pn−1(cos θ) ,

 

n + 1

2n − 1

with the starting values

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

√

 

 

 

 

 

 

 

 

 

P0(cos θ) =

2

 

 

 

 

 

 

 

 

 

 

 

 

,

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P1(cos θ) =

3

cos θ.

 

 

 

 

 

 

2

 

A.2 Legendre Functions

259

(2) For m ≥ 1, compute Pnm+1 by using the recurrence relation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2n + 1)(2n + 3)

cos θP m(cos θ)

 

 

 

 

P m

(cos θ) =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(n + 1 − m)(n + 1 + m)

 

 

 

 

n+1

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2n + 3)(n − m)(n + m)

P m

(cos θ),

 

 

 

 

−

 

 

(2n

n

≥

m

 

 

 

−

1)(n + 1

−

m)(n + 1 + m)

 

n−1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

with the initial values

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pmm−1(cos θ) = 0,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2m + 1

 

 

 

 

 

 

 

 

 

 

Pmm(cos θ) =

 

(2m − 1)!! sinm θ.

 

 

 

 

 

 

2 (2m)!

 

 

 

Considering the angular functions πnm, we see that πn0 diverges at θ = 0 and θ = π. Because in our applications, the product mπnm appears explicitly, we set πn0 = 0 for n ≥ 0. For m ≥ 1, the angular functions πnm are computed by using (A.19), (A.21) and (A.22), and the recurrence relations for the normalized associated Legendre functions.

The angular functions τnm can be calculated with the following algorithm.

(1) For m = 0, compute

 

 

 

 

d

P

 

(cos θ) = P

(cos θ)

 

 

 

 

 

 

 

 

 

n

 

 

 

 

 

d cos θ

 

 

 

 

 

 

 

 

n

 

 

 

 

 

by using the recurrence relation

 

 

 

 

 

 

 

 

(cos θ) = n

 

 

 

P

 

 

 

(cos θ) +

 

 

 

cos θP

 

P

 

2n + 1

 

 

 

 

2n + 1

 

(cos θ)

 

 

 

 

n−1

2n − 1

 

n

 

 

2n − 1

 

 

 

n−1

 

with the starting value

P0 (cos θ) = 0,

and set [144]

τ 0

(θ) =

−

sin θP

(cos θ) , n

≥

0 .

n

 

n

 

 

(2) For m ≥ 1, compute

τnm(θ) = n cos θπnm (θ)

τ m with the recurrence relation

 

 

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

−

(n + m)

 

 

(2n + 1)(n − m)

πm

(θ),

n

≥

m.

 

 

 

 

 

(2n

−

1)(n + m)

 

n−1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Смотрите также файлы