Fundamentals of the Physics of Solids / 16-back-matter

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Российский химико-технологический университет им. Д.И. Менделеева

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2l + 1 (l − m)! .

Plm(x)Plm(x) dx = δl,l

2l + 1 (l − m)! .

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618 C Mathematical Formulas

ζ(3/2) = 2.612 ,		ζ(5/2)	= 1.341 .	(C.3.13)
When the sum runs over odd numbers only,
∞	1
		= (1 − 2−n)ζ(n) .		(C.3.14)
	(2k + 1)n	= (1 − 2−n)ζ(n) .		(C.3.14)

k=0

On the other hand, when the terms are added with an alternating sign, the

Dirichlet eta function or alternating zeta function is obtained:

∞ ( 1)k−1

η(n) =

−

= (1 − 21−n)ζ(n) .

(C.3.15)

k=1

Its integral representation can be analytically continued as

∞

−

∞ ∞

(−1)k−1e−kxxz−1 dx .

η(z) =

dx =

k=1

(C.3.16)

Γ (z)

ex + 1

Γ (z)

For particular values of the argument

η(1) = ln 2 ,

η(2) =

π2

η(4) =

7π4

(C.3.17)

720

The Bernoulli Numbers and their Relationship with the Zeta

Function

The Bernoulli numbers Bn are deﬁned via the series expansion

∞

(C.3.18)

−

n=0

The ﬁrst few Bernoulli numbers are

B0 = 1 , B1 = −

, B2 =

, B2n+1 = 0 ,

(C.3.19)

B4 = −

, B6 =

, B8 = −

, B10

For positive even (and negative odd) numbers the ζ function can also be

expressed in terms of the Bernoulli numbers:
ζ(2n) = (−1)n−1		(2π)2n					(C.3.20)
						B2n
		2(2n)!
and			B2n
ζ(1 − 2n) = −							(C.3.21)
					,
			2n
or conversely	(−1)n−12(2n)!
B2n =				ζ(2n) .			(C.3.22)
	(2π)2n

C.3 Special Functions

619

The Gamma Function

The gamma function (also called Euler’s gamma function or the factorial function) can be deﬁned via the recursion relation

Γ (z + 1) = zΓ (z) ,

(C.3.23)

with the additional condition Γ (1) = 1. In the Re z > 0 region, when the condition Re k > 0 is met, it can be represented by the integrals

∞∞

Γ (z) = 0

tz−1e−t dt = kz 0

tz−1e−kt dt .

(C.3.24)

It can be easily shown by integration by parts that these expressions satisfy the above recursion formula. Via analytical continuation they can be extended to the whole complex plane, where it has poles at nonpositive integers.

By iterating the recursion formula for integer values of z,

Γ (n) = (n − 1)! .

(C.3.25)

Therefore the factorial can be deﬁned for noninteger values of z via the gamma function. For z = 12

∞∞

Γ ( 21 ) = 0	t−1/2e−t dt = 2	0	e−t2	dt = √		,	(C.3.26)
					π

thus for half-integer values of z

Γ (n + 1 ) =

(2n − 1)!!

√

Consequently 21 ! = Γ ( 23 ) = 21 √

= 0.886.

For large values of the argument,

Γ (x) = /

xx e−x 1 +

12x

+ 288x2 + . . . .

2π

This leads to the Stirling formula:

√

n! = nΓ (n)

2πn n

e−

≈

The digamma function (or psi function) is deﬁned as

ψ(z) = ddz ln Γ (z) .

(C.3.27)

(C.3.28)

(C.3.29)

(C.3.30)

Apart from negative integers, the function can be given in the form of an asymptotic series,

620 C Mathematical Formulas

			∞	1	1	(C.3.31)
ψ(z + 1) = −γ − n=1 z + n					− n ,	(C.3.31)

where γ is the Euler–Mascheroni constant (or Euler’s constant):
	n	1
		1
γ = −ψ(1) = n→∞ #		k − ln n$		≈ 0.577 215 . . . .		(C.3.32)
lim	k=1
lim

For z = 1
2
ψ( 21 ) = −γ − 2 ln 2 .						(C.3.33)

C.3.3 Bessel Functions

The Bessel functions (also called cylinder functions) of order ν are deﬁned as solutions to the Bessel di erential equation

z2	d2w(z)		+ z	dw(z)	+ (z2 − ν2)w(z) = 0 ,	(C.3.34)
	dz2			dz

where ν can be any real or complex number. One particular solution can be written as a power series of z,

∞	( 1)k	z	2k+ν	(C.3.35)
Jν (z) = k=0 k!Γ (ν−+ k + 1)		2	.	(C.3.35)

Except for integer orders ν = n, when
J−n(z) = (−1)nJn(z) ,				(C.3.36)

Jν (z) and J−ν (z) are linearly independent, and the general solution of the Bessel di erential equation is

Zν (z) = c1Jν (z) + c2J−ν (z) .

(C.3.37)

There are two classes of solutions to the Bessel equation for any ν with di erent analytic properties. The functions Jν (z) are called Bessel functions of the ﬁrst kind (or simply Bessel functions), while the Bessel functions of the second kind (more commonly called Neumann functions or Weber functions) can be written as

Nν (z) =	Jν (z) cos(νπ) − J−ν (z)	.	(C.3.38)
	sin(νπ)

For integer orders n the Neumann function Nn(z) is deﬁned as the ν → n limit of Nν (z). The Bessel functions of the third kind, also known as Hankel functions are special combinations of the Bessel functions of the ﬁrst and second kinds:

					C.3 Special Functions 621
Hν(1)(z) = Jν (z) + iNν (z) ,			Hν(2)(z) = Jν (z) − iNν (z) .			(C.3.39)
For large values of \|z\|

Jν (z) /	πz		cos	z − 21 πν − 41 π ,		(C.3.40-a)
		2

Nν (z) /		2			21 πν − 41 π	(C.3.40-b)
Nν (z) /		πz sin		z −	21 πν − 41 π	(C.3.40-b)

asymptotically.

The Bessel functions of integer order have a simple integral representation:

	i−n	π		i−n	2π
Jn(z) =		0	eiz cos θ cos(nθ) dθ =		0	eiz cos θ einθ dθ .	(C.3.41)
	π			2π

Among the Bessel functions of fractional order particularly important are the Bessel functions of half-integer order. Instead of Jn+1/2(z), it is customary

to use the spherical Bessel function of the ﬁrst kind

jn(z) = /

Jn+1/2(z) ,

(C.3.42)

which satisﬁes the equation

d2jn(z)

djn(z)

+ 2z

+ [z2 − n(n + 1)]jn(z) = 0 ,

(C.3.43)

dz2

where n = 0, ±1, ±2, . . . . It can be shown that

jn(z) = zn −

1 d

n sin z

(C.3.44)

The explicit expressions for the ﬁrst few spherical Bessel functions are

j0(z) =

sin z

j1(z) =

sin z

−

cos z

(C.3.45)

j2(z) =

−

sin z −

cos z .

For small values of z the leading-order contribution is

jn(z)

(C.3.46)

(2n + 1)!!

while for large values of z

622 C Mathematical Formulas

1		cos		i−(n+1)	eiz + (−1)n+1e−iz
jn(z)			z − 21 (n + 1)π			(C.3.47)
	z			2z

asymptotically.

Just like the Bessel di erential equation, (C.3.43) also has other sets of solutions, the spherical Neumann functions or spherical Bessel functions of the second kind :

nn(z) =	/		/		(C.3.48)
		2z Nn+1/2(z) = (−1)n+1		2z J−n−1/2(z) ,
		π		π

and the spherical Hankel functions or spherical Bessel functions of the third kind :

h(1)n (z) = jn(z) + inn(z) , h(2)n (z) = jn(z) − inn(z) . (C.3.49)

In analogy with (C.3.44), the spherical Bessel functions of the second kind may be written as

nn(z) = −zn −	1 d		n cos z			(C.3.50)
					.
	z	dz		z

The ﬁrst few spherical Bessel functions of the second and third kinds are

n0(z) = −	cos z	,
	z
n1(z) = −	cos z	−	sin z	,	(C.3.51-a)
	z2		z

n2(z) = −	3	1		3		sin z ,
		−		cos z −
	z3		z		z2

h(1)0 (z) = −zi eiz ,

						i
(1)	(z) = eiz	1			−	i			(C.3.51-b)
h1	(z) = eiz	−		z	−	z2	,		(C.3.51-b)
h2	(z) = eiz	z − z2 − z3 .
(1)			i		3			3i

For small values of z the leading-order singularity of nn(z) is of the type

(z)

(2n − 1)!!

(C.3.52)

zn+1

while for large values of z

nn(z)

sin

1 (n + 1)π

(C.3.53-a)

−

h(1)

(z)

ei[z−(n+1)π/2]

(C.3.53-b)

asymptotically.

C.4 Orthogonal Polynomials

623

The modiﬁed Bessel functions of order ν satisfy the modiﬁed Bessel differential equation

z2 d2w(z) dz2

+ z	dw(z)	− (z2 + ν2)w(z) = 0 .	(C.3.54)
	dz

Comparison with the Bessel di erential equation immediately shows that the series expansion in powers of z,

∞	1	z	ν+2k
Iν (z) = k=0 k!Γ (ν + k + 1)		2	(C.3.55)

is a solution. This is called modiﬁed Bessel function of the ﬁrst kind. For integer orders ν = n

In(z) = i−nJn(iz) ,

(C.3.56)

while for general ν the solutions Iν (z) and I−ν (z) are linearly independent. A particular linear combination of these functions,

Kν (z) =	π	I−ν (z) − Iν (z)	(C.3.57)
		sin(νz)
2

is the modiﬁed Bessel function of the second kind or Macdonald function. It can also be obtained from the analytical continuation of the Hankel function:

Kν (z) = 1 iπeiπν/2H(1)		(iz) .	(C.3.58)
2	ν

Of special importance is the function large values of its argument are

− ln(

K0(z) = / π e

K0; its asymptotic form for small and

12 z) + γ	z	1	,
−z	z	1	(C.3.59)
			,

where γ = 0.57721 is the Euler–Mascheroni constant given in (C.3.32).

C.4 Orthogonal Polynomials

C.4.1 Hermite Polynomials

The Hermite polynomials are the solutions of the di erential equation

d2Hn(x)	− 2x	dHn(x)	+ 2nHn(x) = 0 , n = 0, 1, 2, . . . .	(C.4.1)
dx2		dx

A compact representation is given by Rodrigues’ formula:

624 C Mathematical Formulas

Hn(x) = (−1)n ex2 dn e−x2 dxn

They are orthogonal with respect to the weight function w(x) = e−x

∞

e−x2 Hn(x)Hm(x) dx = 2nn!√πδnm .

−∞

(C.4.2)

(C.4.3)

The explicit expressions for the ﬁrst few Hermite polynomials are

H0(x) = 1 , H1(x) = 2x ,

H2(x) = 4x2 − 1 , (C.4.4)

H3(x) = 8x3 − 12x ,

H4(x) = 16x4 − 48x2 + 12 , while their general formula is


[n/2]		n!(2x)n−2k
		n!(2x)n−2k
	(−1)k k!(n		−	2k)! .	(C.4.5)
Hn(x) =	(−1)k k!(n			2k)! .	(C.4.5)

k=0

C.4.2 Laguerre Polynomials

The Laguerre polynomials are the solutions of the di erential equation

d2Ln(x)

+ (1 − x)

dLn(x)

+ nLn(x) = 0

n = 0, 1, 2, . . . .

(C.4.6)

dx2

Alternatively, they can be given by Rodrigues’ formula:

e−xxn .

Ln(x) =

(C.4.7)

dxn

They are orthogonal with respect to the weight function w(x) = e−x:

∞

e−xLm(x)Ln(x) dx = δm,n .

(C.4.8)

The explicit expressions for the ﬁrst few Laguerre polynomials are

L0(x) = 1 ,

L1(x) = 1 − x ,

L2(x) = 1 − 2x +

x2 ,

(C.4.9)

L3(x) = 1 − 3x +

x2 −

x3 .

C.4 Orthogonal Polynomials

625

The generalized Laguerre polynomials are the solutions of the di erential equation

x	d2Ln(α)(x)	+ (α + 1 − x)	dLn(α)(x)		+ nLn(α)(x) = 0	(C.4.10)
	dx2		dx

for nonnegative values of n, where α is an arbitrary complex number. Consequently, Rodrigues’ formula reads

	1		dn
Ln(α)(x) =		exx−α		e−xxn+α .	(C.4.11)
	n!		dxn

For α > −1 the generalized Laguerre polynomials satisfy the orthogonality relation with respect to the weight function w(x) = e−xxα

∞	e−xxαLn(α)(x)Ln	(x) dx =	n!	δnn .	(C.4.12)
0
	(α)		Γ (α + n + 1)

Of particular interest are the generalized Laguerre polynomials whose index α is a nonnegative integer. Their polynomial form is

n	n + m			xk
	n + m			xk
Lnm(x) = k=0(−1)k n			k		,	(C.4.13)
		−

while their connection with ordinary Laguerre polynomials is given by

dm
Lnm(x) = (−1)m dxm Ln+m(x)	(C.4.14)

and L0n(x) = Ln(x).

C.4.3 Legendre Polynomials

The Legendre polynomials are deﬁned on the interval |x| ≤ 1, and satisfy the di erential equation

(1 − x2)

d2Pl(x)

− 2x

dPl(x)

+ l(l + 1)Pl(x) = 0 ,

(C.4.15)

dx2

that is,

Pl(x) =

1 dl

1 [l/2]

−l

(C.4.16)

2ll! dxl (x2 − 1)l =

k=0(−1)k k

xl−2k .

The Legendre polynomials are mutually orthogonal, however they are not normalized:

626 C Mathematical Formulas

+1		2l + 1 .	(C.4.17)
	Pl(x)Pl (x) dx = δl,l	2l + 1 .	(C.4.17)
		2
−1

The explicit expressions of the ﬁrst few polynomials are

P0(x) = 1 ,
P1	(x) = x = cos θ ,
		1	(3x2	1			(3 cos 2θ + 1) ,
P2	(x) =			− 1) =							(C.4.18)
		2			4
P3	(x) =	1	(5x3	− 3x) =		1		(5 cos 3θ + 3 cos θ) ,
		2					8
		1				1
P4	(x) =		(35x4 − 30x2 + 3) =							(35 cos 4θ + 20 cos 2θ + 9) .
		8							64

The same equation, (C.4.15), is satisﬁed by the Legendre polynomials of the second kind,

Q0(x) =

Q1(x) =

Q2(x) =

1 ln		1 + x				,
1 ln		1 − x				,
2

x			1	+ x	− 1 ,
	ln								(C.4.19)
2	ln		1	− x					(C.4.19)
3x2			1			1 + x		3
	−			ln			−		x .
	4			ln		1 − x	−	2	x .

The associated Legendre polynomials are solutions of the equation

(1 − x )	d2P m(x)		dP m(x)		+	l(l + 1) −	m2	Pl	(x) = 0 ,	(C.4.20)
	dx2	− 2x		dx			1 − x2
2	l			l				m
that is
Plm(x) = (1 − x2)m/2				dmPl(x)		l, m = 0, 1, 2, . . .			m ≤ l ,	(C.4.21)
				dxm

while the associated Legendre polynomials of the second kind are deﬁned as

Qlm(x) = (1 − x2)m/2	dmQl(x)	l, m = 0, 1, 2, . . . m ≤ l . (C.4.22)
	dxm

In several references an additional factor (−1)m appears in the deﬁnition. The associated Legendre polynomials satisfy the orthogonality relation

The explicit expressions for the ﬁrst few associated Legendre polynomials are

C.4

Orthogonal Polynomials 627

P11(x) = (1 − x2)1/2 = sin θ ,

P21

(x) = 3(1 − x2)1/2x =

sin 2θ ,

− cos 2θ) ,

P22

(x) = 3(1 − x2) =

(C.4.24)

(1 − x2)1/2(5x2 − 1) =

(sin θ + 5 sin 3θ) ,

P31

(x) =

(cos θ − cos 3θ) ,

P32

(x) = 15(1 − x2)x =

P33

(x) = 15(1 − x2)3/2 =

(3 sin θ − sin 3θ) .

C.4.4 Spherical Harmonics

In physics, instead of the associated Legendre polynomials, the functions

m+|m|

1/2

(2l + 1)(l

m )!

1/2

−

2π

2(l + |m|)!

Y m(θ, ϕ) = (

1) 2

− |

P | |(cos θ)eimϕ

(C.4.25) are used, where −l ≤ m ≤ l. They are called spherical harmonics (or surface harmonics). Spherical harmonics satisfy the spherical harmonic di erential equation, which is given by the angular part of Laplace’s equation in spherical coordinates. It is immediately seen from the above form that

−m(θ, ϕ) = ( 1)mY m (θ, ϕ) .			(C.4.26)
Yl	−	l

The spherical harmonics are normalized in such a way that the orthogonality relation

2π π

Y	m		m	(θ, ϕ) sin θ dθ dϕ = δll δmm	(C.4.27)
	l	(θ, ϕ)Y	l

0 0

and the completeness relation

∞ +l		δ(θ − θ )δ(ϕ − ϕ )	(C.4.28)
Y m (θ, ϕ)Y m(θ , ϕ ) =		δ(θ − θ )δ(ϕ − ϕ )	(C.4.28)

l	l
l=0 m=−l		sin θ
l=0 m=−l

= δ(cos θ − cos θ )δ(ϕ − ϕ ) ≡ δ(Ω − Ω )

are both satisﬁed.

According to the addition theorem for spherical harmonics, the product of two spherical harmonics can be written as the linear combination of spherical harmonics:

Yl1	1 (θ, ϕ)Yl2	2 (θ, ϕ) =	7	4π(2l + 1)	(C.4.29)
		l1+l2	l	(2l1 + 1)(2l2 + 1)
m	m	\|		(2l1 + 1)(2l2 + 1)
		\|
		l= l1 −l2\| m=−l

×(l1l2m1m2|lm)(l1l200|l0)Ylm(θ, ϕ) .

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