Integrals with modified Bessel functions.. Airy functions of the first and the second kinds.. Degenerate Hypergeometric Functions Kummer Functions 18.9.1.. Table 18.1 presents some speci
Trang 118.7.2-2 Asymptotic expansions as x → ∞.
I ν (x) = e
x
√
2πx
1+
M
m=1
(–1)m(4ν2–1)(4ν2–32) [4ν2– (2m–1)2]
m! (8x)m
,
K ν (x) = 2π x e
–x
1+
M
m=1
(4ν2–1)(4ν2–32) [4ν2– (2m–1)2]
m! (8x)m
The terms of the order of O(x–M–1) are omitted in the braces
18.7.2-3 Integrals with modified Bessel functions
x
0
x λ I
ν (x) dx = x
λ+ν+1
2ν (λ + ν +1 )Γ(ν +1 )F
λ + ν +1
λ + ν +3
2 , ν +1;
x2
4
, Re(λ+ν) > –1 ,
where F (a, b, c; x) is the hypergeometric series (see Subsection 18.10.1),
x
0 x
λ K
ν (x) dx = 2ν–1Γ(ν)
λ – ν +1x λ–ν+1F
λ – ν +1
2 , 1– ν,
λ – ν +3
x2 4
+ 2–ν–1Γ(–ν)
λ + ν +1 x λ+ν+1F
λ + ν +1
2 , 1+ ν,
λ + ν +3
x2 4
, Re λ >|Re ν|–1
18.8 Airy Functions
18.8.1 Definition and Basic Formulas
18.8.1-1 Airy functions of the first and the second kinds
The Airy function of the first kind, Ai(x), and the Airy function of the second kind, Bi(x),
are solutions of the Airy equation
y
xx – xy =0
and are defined by the formulas
Ai(x) = 1
π
∞
0 cos
1
3t3+ xt
dt,
Bi(x) = 1
π
∞ 0
exp –13t3+ xt
+ sin 13t3+ xt
dt
Wronskian: W{Ai(x), Bi(x)}=1/π
18.8.1-2 Relation to the Bessel functions and the modified Bessel functions
Ai(x) = 13√
x
I–1 3(z) – I1 3(z)
= π–1
1
3x K1 3(z), z= 23x3 2,
Ai(–x) = 13√
x
J–1 3(z) + J1 3(z)
,
Bi(x) =
1
3x
I–1 3(z) + I1 3(z)
,
Bi(–x) =
1
3x
J–1 3(z) – J1 3(z)
Trang 2
18.8.2 Power Series and Asymptotic Expansions
18.8.2-1 Power series expansions as x →0
Ai(x) = c1f (x) – c2g (x), Bi(x) = √
3[c1f (x) + c2g (x)],
f (x) =1+ 1
3!x
3+ 1 × 4
6! x
6+ 1 × 4 × 7
9! x
9+· · · =∞
k=0
3k 1 3
k
x3k
(3k)!,
g (x) = x + 2
4!x
4+ 2 × 5
7! x
7+ 2 × 5 × 8
10! x
10+· · · =∞
k=0
3k 2 3
k
x3k+1
(3k+1)!,
where c1=3– 2 3/Γ(2/3)≈ 0.3550 and c2=3– 1 3/Γ(1/3) ≈ 0.2588
18.8.2-2 Asymptotic expansions as x → ∞.
For large values of x, the leading terms of asymptotic expansions of the Airy functions are
Ai(x) 12π–1 2x– 1 4exp(–z), z= 2
3x3 2,
Ai(–x) π– 1 2x– 1 4sin z+ π
4
,
Bi(x) π– 1 2x– 1 4exp(z),
Bi(–x) π– 1 2x– 1 4cos z+ π
4
18.9 Degenerate Hypergeometric Functions (Kummer
Functions)
18.9.1 Definitions and Basic Formulas
18.9.1-1 Degenerate hypergeometric functionsΦ(a, b; x) and Ψ(a, b; x).
The degenerate hypergeometric functions (Kummer functions) Φ(a, b; x) and Ψ(a, b; x) are
solutions of the degenerate hypergeometric equation
xy
xx + (b – x)y x – ay =0
In the case b≠ 0, –1, –2, –3, , the function Φ(a, b; x) can be represented as Kummer’s
series:
Φ(a, b; x) =1+
∞
k=1
(a) k (b) k
x k
k!,
where (a) k = a(a +1) (a + k –1), (a)0 =1
Table 18.1 presents some special cases whereΦ can be expressed in terms of simpler
functions
The functionΨ(a, b; x) is defined as follows:
Ψ(a, b; x) = Γ(a – b +Γ(1– b)1
)Φ(a, b; x) + Γ(b – Γ(a)1)x1 –b Φ(a – b +1, 2– b; x).
Table 18.2 presents some special cases whereΨ can be expressed in terms of simpler
functions
Trang 3TABLE 18.1 Special cases of the Kummer functionΦ(a, b; z)
x e sinh x
Incomplete gamma function
γ (a, x) =
x
0
e–t t a–1 dt
1
2
3
2 –x2
√ π
2 erf x
Error function
erf x = √2
π
x
0 exp(–t2) dt
2
2
n! ( 2n )!
– 1 2
–n
H2n(x) Hermite polynomials
H n (x) = (–1 )n e 2 d
dx n e
–x2 ,
n= 0 , 1 , 2,
2
2
n! ( 2n + 1 )!
– 1 2
–n
H2n+1(x)
(b) n L
(b–1)
n (x)
Laguerre polynomials
L(n α) (x) = e x
–α
n!
d
dx n e
–x x n+α ,
α = b–1 ,
(b) n = b(b+1) (b+n–1 )
ν+ 1
2 2ν+1 2x Γ(1+ν)e x
x 2
–ν
I (x)
Modified Bessel functions
I (x)
n+ 1 2n + 2 2x Γn+ 3
2
e
x 2
–n–1
I n+1(x)
18.9.1-2 Kummer transformation and linear relations
Kummer transformation:
Φ(a, b; x) = e x Φ(b – a, b; –x), Ψ(a, b; x) = x1–bΨ(1+ a – b,2– b; x).
Linear relations forΦ:
(b – a) Φ(a –1, b; x) + (2a – b + x) Φ(a, b; x) – aΦ(a +1, b; x) =0,
b (b –1)Φ(a, b –1; x) – b(b –1+ x) Φ(a, b; x) + (b – a)xΦ(a, b +1; x) =0,
(a – b +1)Φ(a, b; x) – aΦ(a +1, b; x) + (b –1)Φ(a, b –1; x) =0,
b Φ(a, b; x) – bΦ(a –1, b; x) – x Φ(a, b +1; x) =0,
b (a + x) Φ(a, b; x) – (b – a)xΦ(a, b +1; x) – ab Φ(a +1, b; x) =0,
(a –1+ x) Φ(a, b; x) + (b – a)Φ(a –1, b; x) – (b –1)Φ(a, b –1; x) =0
Linear relations forΨ:
Ψ(a –1, b; x) – (2a – b + x) Ψ(a, b; x) + a(a – b +1)Ψ(a +1, b; x) =0,
(b – a –1)Ψ(a, b –1; x) – (b –1+ x) Ψ(a, b; x) + xΨ(a, b +1; x) =0,
Ψ(a, b; x) – aΨ(a +1, b; x) – Ψ(a, b –1; x) =0,
(b – a) Ψ(a, b; x) – xΨ(a, b +1; x) + Ψ(a –1, b; x) =0,
(a + x) Ψ(a, b; x) + a(b – a –1)Ψ(a +1, b; x) – x Ψ(a, b +1; x) =0,
(a –1+ x) Ψ(a, b; x) – Ψ(a –1, b; x) + (a – c +1)Ψ(a, b –1; x) =0
Trang 4TABLE 18.2 Special cases of the Kummer functionΨ(a, b; z)
Incomplete gamma function
Γ(a, x) =
∞
x
e–t t a–1 dt
1
2
1
√
π exp(x2) erfc x
Complementary error function
erfc x = 2
√ π
∞
x
exp(–t2) dt
Exponential integral
Ei(x) =
x
–∞
e t
t dt
Logarithmic integral
li x =
x
0
dt t
1
2–
n
2
3
2 x2 2–n x–1H n (x)
Hermite polynomials
H n (x) = (–1 )n e 2 d
dx n e
–x2 ,
n= 0 , 1 , 2,
ν+ 1
2 2ν+1 2x π–1/2(2x)–ν x K ν (x) Modified Bessel functions
K ν (x)
18.9.1-3 Differentiation formulas and Wronskian
Differentiation formulas:
d
dx Φ(a, b; x) = a
b Φ(a +1, b +1; x), d
dx Ψ(a, b; x) = –aΨ(a +1, b +1; x),
d n
dx n Φ(a, b; x) = (a) n
(b) n Φ(a + n, b + n; x),
d n
dx n Ψ(a, b; x) = (–1)n (a) n Ψ(a + n, b + n; x).
Wronskian:
W(Φ, Ψ) = ΦΨ x–Φ xΨ = –Γ(a) Γ(b) x–b e x.
18.9.1-4 Degenerate hypergeometric functions for n =0,1,2,
Ψ(a, n +1; x) = (–1)n–1
n!Γ(a – n)
Φ(a, n+1; x) ln x
+
∞
r=0
(a) r
(n +1)r
ψ (a + r) – ψ(1+ r) – ψ(1+ n + r) x r
r!
+ (n –1)!
Γ(a)
n–1
r=0
(a – n) r
(1– n) r
x r–n
r! ,
where n =0, 1, 2, (the last sum is dropped for n =0), ψ(z) = [ln Γ(z)]
zis the logarithmic
derivative of the gamma function,
ψ(1) = –C, ψ(n) = –C +
n–1
k=1
k–1,
whereC =0.5772 . is the Euler constant
Trang 5If b <0, then the formula
Ψ(a, b; x) = x1–b Ψ(a – b +1, 2– b; x)
is valid for any x.
For b≠ 0, –1, –2, –3, , the general solution of the degenerate hypergeometric equation
can be represented in the form
y = C1Φ(a, b; x) + C2Ψ(a, b; x),
and for b =0, –1, –2, –3, , in the form
y = x1–b
C1Φ(a – b +1, 2– b; x) + C2Ψ(a – b +1, 2– b; x)
18.9.2 Integral Representations and Asymptotic Expansions
18.9.2-1 Integral representations
Φ(a, b; x) = Γ(a) Γ(b – a) Γ(b)
1
0 e
xt t a–1(1– t) b–a–1dt (for b > a >0),
Ψ(a, b; x) = 1
Γ(a)
∞
0 e –xt t a–1(1+ t) b–a–1dt (for a >0, x >0), whereΓ(a) is the gamma function.
18.9.2-2 Asymptotic expansion as|x|→ ∞.
Φ(a, b; x) = Γ(a) Γ(b) e x x a–bN
n=0
(b – a) n(1– a) n
–n + ε
, x>0,
Φ(a, b; x) = Γ(b – a) Γ(b) (–x)–a
N
n=0
(a) n (a – b +1)n
n! (–x)
–n + ε
, x<0,
Ψ(a, b; x) = x–aN
n=0
(–1)n (a) n (a – b +1)n
–n + ε
, –∞ < x < ∞, where ε = O(x–N–1)
18.9.2-3 Integrals with degenerate hypergeometric functions
Φ(a, b; x) dx = b–1
a–1Ψ(a –1, b –1; x) + C,
Ψ(a, b; x) dx = 11
– a Ψ(a –1, b –1; x) + C,
x n Φ(a, b; x) dx = n! n+
1
k=1
(–1)k+1(1– b) k x n–k+1
(1– a) k (n – k +1)! Φ(a – k, b – k; x) + C,
x n Ψ(a, b; x) dx = n! n+
1
k=1
(–1)k+1x n–k+1
(1– a) k (n – k +1)!Ψ(a – k, b – k; x) + C.
Trang 618.9.3 Whittaker Functions
The Whittaker functions M k,μ (x) and W k,μ (x) are linearly independent solutions of the
Whittaker equation:
y
xx+
–14 + 12k+ 14 – μ2
x–2
y=0 The Whittaker functions are expressed in terms of degenerate hypergeometric functions as
M k,μ (x) = x μ+1 2e–x/2Φ 12 + μ – k,1+2μ ; x
,
W k,μ (x) = x μ+1 2e–x/2Ψ 12 + μ – k,1+2μ ; x
18.10 Hypergeometric Functions
18.10.1 Various Representations of the Hypergeometric Function
18.10.1-1 Representations of the hypergeometric function via hypergeometric series
The hypergeometric function F (α, β, γ; x) is a solution of the Gaussian hypergeometric
equation
x (x –1)y xx + [(α + β +1)x – γ]y x + αβy =0
For γ ≠ 0, –1, –2, –3, , the function F (α, β, γ; x) can be expressed in terms of the
hypergeometric series:
F (α, β, γ; x) =1+
∞
k=1
(α) k (β) k (γ) k
x k
k!, (α) k = α(α +1) (α + k –1), which certainly converges for|x|<1
If γ is not an integer, then the general solution of the hypergeometric equation can be
written in the form
y = C1F (α, β, γ; x) + C2x1 –γ F (α – γ +1, β – γ +1, 2– γ; x).
Table 18.3 shows some special cases where F can be expressed in term of elementary
functions
18.10.1-2 Integral representation
For γ > β >0, the hypergeometric function can be expressed in terms of a definite integral:
F (α, β, γ; x) = Γ(γ)
Γ(β) Γ(γ – β)
1
0 t
β–1(1– t) γ–β–1(1– tx)–α dt, whereΓ(β) is the gamma function.
18.10.2 Basic Properties
18.10.2-1 Linear transformation formulas
F (α, β, γ; x) = F (β, α, γ; x),
F (α, β, γ; x) = (1– x) γ–α–β F (γ – α, γ – β, γ; x),
F (α, β, γ; x) = (1– x)–α F
α , γ – β, γ; x
x–1
,
F (α, β, γ; x) = (1– x)–β F
β , γ – α, γ; x
x–1
Trang 7
TABLE 18.3
Some special cases where the hypergeometric function F (α, β, γ; z)
can be expressed in terms of elementary functions.
n
k=0
(–n) k (β) k (γ) k
x k
k!, where n =1 , 2,
n
k=0
(–n) k (β) k (–n – m) k
x k
k!, where n =1 , 2,
1 +√
1– x
2
–2α
1– x
1 +√
1– x
2
1– 2α
α α+12 32 x2 (1+ x)1– 2α – ( 1– x)1– 2α
2x ( 1 – 2α )
2
( 1+ x)–2α+ ( 1– x)–2α
1– x 2– 2α
(α –1 ) sin( 2x )
α 1– α 12 –x2 1+ x2+ x 2α–1
+ 1+ x2– x 2α–1
2√1+ x2
(α –1 ) sin( 2x )
cos x
2 1+ x2+ x 2α
+ 1+ x2– x 2α
x ln(x +1 )
1
2xln
1+ x
1– x
1
x arctan x
1
x arcsin x
1
x arcsinh x
n+ 1 n + m +1 n + m + l +2 x
(– 1 )m (n + m + l +1 )!
n ! l! (n + m)! (m + l)!
d n+m
dx n+m
( 1– x) m+l d
l F
dx l
,
F = –ln(1– x)
x , n , m, l =0 , 1 , 2,