948 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.6.1-2. Some formulas. 2νZ ν (x)=x[Z ν–1 (x)+Z ν+1 (x)], d dx Z ν (x)= 1 2 [Z ν–1 (x)–Z ν+1 (x)] = ν x Z ν (x)–Z ν 1 (x) , d dx [x ν Z ν (x)] = x ν Z ν–1 (x), d dx [x –ν Z ν (x)]=–x –ν Z ν+1 (x), 1 x d dx n [x ν J ν (x)] = x ν–n J ν–n (x), 1 x d dx n [x –ν J ν (x)]=(–1) n x –ν–n J ν+n (x), J –n (x)=(–1) n J n (x), Y –n (x)=(–1) n Y n (x), n = 0, 1, 2, 18.6.1-3. Bessel functions for ν = n 1 2 ,wheren = 0, 1, 2, J 1/2 (x)= 2 πx sin x, J 3/2 (x)= 2 πx 1 x sin x –cosx , J –1/2 (x)= 2 πx cos x, J –3/2 (x)= 2 πx – 1 x cos x –sinx , J n+1/2 (x)= 2 πx sin x – nπ 2 [n/2] k=0 (–1) k (n + 2k)! (2k)! (n – 2k)! (2x) 2k +cos x – nπ 2 [(n–1)/2] k=0 (–1) k (n + 2k + 1)! (2k + 1)! (n – 2k – 1)! (2x) 2k+1 , J –n–1/2 (x)= 2 πx cos x + nπ 2 [n/2] k=0 (–1) k (n + 2k)! (2k)! (n – 2k)! (2x) 2k –sin x + nπ 2 [(n–1)/2] k=0 (–1) k (n + 2k + 1)! (2k + 1)! (n – 2k – 1)! (2x) 2k+1 , Y 1/2 (x)=– 2 πx cos x, Y n+1/2 (x)=(–1) n+1 J –n–1/2 (x), Y –1/2 (x)= 2 πx sin x, Y –n–1/2 (x)=(–1) n J n+1/2 (x), where [A] is the integer part of the number A. 18.6.1-4. Bessel functions for ν = n,wheren = 0, 1, 2, Let ν = n be an arbitrary integer. The relations J –n (x)=(–1) n J n (x), Y –n (x)=(–1) n Y n (x) are valid. The function J n (x)isgivenbythefirst formula in (18.6.1.1) with ν = n,and Y n (x) can be obtained from the second formula in (18.6.1.1) by proceeding to the limit 18.6. BESSEL FUNCTIONS (CYLINDRICAL FUNCTIONS) 949 ν → n. For nonnegative n, Y n (x) can be represented in the form Y n (x)= 2 π J n (x)ln x 2 – 1 π n–1 k=0 (n – k – 1)! k! 2 x n–2k – 1 π ∞ k=0 (–1) k x 2 n+2k ψ(k + 1)+ψ(n + k + 1) k!(n + k)! , where ψ(1)=–C, ψ(n)=–C + n–1 k=1 k –1 , C = 0.5772 is the Euler constant, and ψ(x)= [ln Γ(x)] x is the logarithmic derivative of the gamma function, also known as the digamma function. 18.6.1-5. Wronskians and similar formulas. W (J ν , J –ν )=– 2 πx sin(πν), W (J ν , Y ν )= 2 πx , J ν (x)J –ν+1 (x)+J –ν (x)J ν–1 (x)= 2 sin(πν) πx , J ν (x)Y ν+1 (x)–J ν+1 (x)Y ν (x)=– 2 πx . Here, the notation W (f, g)=fg x – f x g is used. 18.6.2. Integral Representations and Asymptotic Expansions 18.6.2-1. Integral representations. The functions J ν (x)andY ν (x) can be represented in the form of definite integrals (for x > 0): πJ ν (x)= π 0 cos(x sin θ – νθ) dθ –sinπν ∞ 0 exp(–x sinh t – νt) dt, πY ν (x)= π 0 sin(x sin θ – νθ)dθ – ∞ 0 (e νt + e –νt cos πν) e –x sinh t dt. For |ν| < 1 2 , x > 0, J ν (x)= 2 1+ν x –ν π 1/2 Γ( 1 2 – ν) ∞ 1 sin(xt) dt (t 2 – 1) ν+1/2 , Y ν (x)=– 2 1+ν x –ν π 1/2 Γ( 1 2 – ν) ∞ 1 cos(xt) dt (t 2 – 1) ν+1/2 . For ν >– 1 2 , J ν (x)= 2(x/2) ν π 1/2 Γ( 1 2 + ν) π/2 0 cos(x cos t)sin 2ν tdt (Poisson’s formula). For ν = 0, x > 0, J 0 (x)= 2 π ∞ 0 sin(x cosh t) dt, Y 0 (x)=– 2 π ∞ 0 cos(x cosh t) dt. 950 SPECIAL FUNCTIONS AND THEIR PROPERTIES For integer ν = n = 0, 1, 2, , J n (x)= 1 π π 0 cos(nt – x sin t) dt (Bessel’s formula), J 2n (x)= 2 π π/2 0 cos(x sin t)cos(2nt) dt, J 2n+1 (x)= 2 π π/2 0 sin(x sin t)sin[(2n + 1)t] dt. 18.6.2-2. Asymptotic expansions as |x| →∞. J ν (x)= 2 πx cos 4x – 2νπ – π 4 M–1 m=0 (–1) m (ν, 2m)(2x) –2m + O(|x| –2M ) –sin 4x – 2νπ – π 4 M–1 m=0 (–1) m (ν, 2m + 1)(2x) –2m–1 + O(|x| –2M–1 ) , Y ν (x)= 2 πx sin 4x – 2νπ – π 4 M–1 m=0 (–1) m (ν, 2m)(2x) –2m + O(|x| –2M ) +cos 4x – 2νπ – π 4 M–1 m=0 (–1) m (ν, 2m + 1)(2x) –2m–1 + O(|x| –2M–1 ) , where (ν, m)= 1 2 2m m! (4ν 2 – 1)(4ν 2 – 3 2 ) [4ν 2 –(2m – 1) 2 ]= Γ( 1 2 + ν + m) m! Γ( 1 2 + ν – m) . For nonnegative integer n and large x, √ πx J 2n (x)=(–1) n (cos x +sinx)+O(x –2 ), √ πx J 2n+1 (x)=(–1) n+1 (cos x –sinx)+O(x –2 ). 18.6.2-3. Asymptotic for large ν (ν →∞). J ν (x) 1 √ 2πν ex 2ν ν , Y ν (x) – 2 πν ex 2ν –ν , where x is fixed, J ν (ν) 2 1/3 3 2/3 Γ(2/3) 1 ν 1/3 , Y ν (ν) – 2 1/3 3 1/6 Γ(2/3) 1 ν 1/3 . 18.6.2-4. Integrals with Bessel functions. x 0 x λ J ν (x) dx = x λ+ν+1 2 ν (λ + ν + 1) Γ(ν + 1) F λ + ν + 1 2 , λ + ν + 3 2 , ν +1;– x 2 4 ,Re(λ+ν)>–1, 18.6. BESSEL FUNCTIONS (CYLINDRICAL FUNCTIONS) 951 where F (a, b, c; x) is the hypergeometric series (see Section 18.10.1), x 0 x λ Y ν (x) dx =– cos(νπ)Γ(–ν) 2 ν π(λ + ν + 1) x λ+ν+1 F λ + ν + 1 2 , ν + 1, λ + ν + 3 2 ;– x 2 4 – 2 ν Γ(ν) λ – ν + 1 x λ–ν+1 F λ – ν + 1 2 , 1 – ν, λ – ν + 3 2 ;– x 2 4 ,Reλ > |Re ν| – 1. 18.6.3. Zeros and Orthogonality Properties of Bessel Functions 18.6.3-1. Zeros of Bessel functions. Each of the functions J ν (x)andY ν (x)hasinfinitely many real zeros (for real ν). All zeros are simple, except possibly for the point x = 0. The zeros γ m of J 0 (x), i.e., the roots of the equation J 0 (γ m )=0, are approximately given by γ m = 2.4 + 3.13 (m – 1)(m = 1, 2, ), with a maximum error of 0.2%. 18.6.3-2. Orthogonality properties of Bessel functions. 1 ◦ .Letμ = μ m be positive roots of the Bessel function J ν (μ), where ν >–1 and m = 1, 2, 3, Then the set of functions J ν (μ m r/a) is orthogonal on the interval 0 ≤ r ≤ a with weight r: a 0 J ν μ m r a J ν μ k r a rdr= 0 if m ≠ k, 1 2 a 2 J ν (μ m ) 2 = 1 2 a 2 J 2 ν+1 (μ m )ifm = k. 2 ◦ .Letμ = μ m be positive zeros of the Bessel function derivative J ν (μ), where ν >–1 and m = 1, 2, 3, Then the set of functions J ν (μ m r/a) is orthogonal on the interval 0 ≤ r ≤ a with weight r: a 0 J ν μ m r a J ν μ k r a rdr = ⎧ ⎨ ⎩ 0 if m ≠ k, 1 2 a 2 1 – ν 2 μ 2 m J 2 ν (μ m )ifm = k. 3 ◦ .Letμ = μ m be positive roots of the transcendental equation μJ ν (μ)+sJ ν (μ)=0,where ν >–1 and m = 1, 2, 3, Then the set of functions J ν (μ m r/a) is orthogonal on the interval 0 ≤ r ≤ a with weight r: a 0 J ν μ m r a J ν μ k r a rdr= ⎧ ⎨ ⎩ 0 if m ≠ k, 1 2 a 2 1 + s 2 – ν 2 μ 2 m J 2 ν (μ m )ifm = k. 4 ◦ .Letμ = μ m be positive roots of the transcendental equation J ν (λ m b)Y ν (λ m a)–J ν (λ m a)Y ν (λ m b)=0 (ν >–1, m = 1, 2, 3, ). Then the set of functions Z ν (λ m r)=J ν (λ m r)Y ν (λ m a)–J ν (λ m a)Y ν (λ m r), m = 1, 2, 3, ; 952 SPECIAL FUNCTIONS AND THEIR PROPERTIES satisfying the conditions Z ν (λ m a)=Z ν (λ m b)=0 is orthogonal on the interval a ≤ r ≤ b with weight r: b a Z ν (λ m r)Z ν (λ k r)rdr = ⎧ ⎨ ⎩ 0 if m ≠ k, 2 π 2 λ 2 m J 2 ν (λ m a)–J 2 ν (λ m b) J 2 ν (λ m b) if m = k. 5 ◦ .Letμ = μ m be positive roots of the transcendental equation J ν (λ m b)Y ν (λ m a)–J ν (λ m a)Y ν (λ m b)=0 (ν >–1, m = 1, 2, 3, ). Then the set of functions Z ν (λ m r)=J ν (λ m r)Y ν (λ m a)–J ν (λ m a)Y ν (λ m r), m = 1, 2, 3, ; satisfying the conditions Z ν (λ m a)=Z ν (λ m b)=0 is orthogonal on the interval a ≤ r ≤ b with weight r: b a Z ν (λ m r)Z ν (λ k r)rdr= ⎧ ⎪ ⎨ ⎪ ⎩ 0 if m ≠ k, 2 π 2 λ 2 m 1 – ν 2 b 2 λ 2 m J ν (λ m a) 2 J ν (λ m b) 2 – 1 – ν 2 a 2 λ 2 m if m = k. 18.6.4. Hankel Functions (Bessel Functions of the Third Kind) 18.6.4-1. Definition. The Hankel functions of the first kind and the second kind are related to Bessel functions by H (1) ν (z)=J ν (z)+iY ν (z), H (2) ν (z)=J ν (z)–iY ν (z), where i 2 =–1. 18.6.4-2. Expansions as z → 0 and z →∞. Asymptotics for z → 0: H (1) 0 (z) 2i π ln z, H (1) ν (z) – i π Γ(ν) (z/2) ν (Re ν > 0), H (2) 0 (z) – 2i π ln z, H (2) ν (z) i π Γ(ν) (z/2) ν (Re ν > 0). Asymptotics for |z| →∞: H (1) ν (z) 2 πz exp i z – 1 2 πν – 1 4 π (–π <argz < 2π), H (2) ν (z) 2 πz exp –i z – 1 2 πν – 1 4 π (–2π <argz < π). 18.7. MODIFIED BESSEL FUNCTIONS 953 18.7. Modified Bessel Functions 18.7.1. Definitions. Basic Formulas 18.7.1-1. Modified Bessel functions of the first and the second kind. The modified Bessel functions of the first kind, I ν (x), and the modified Bessel functions of the second kind, K ν (x) (also called the Macdonald function), of order ν are solutions of the modified Bessel equation x 2 y xx + xy x –(x 2 + ν 2 )y = 0 and are defined by the formulas I ν (x)= ∞ k=0 (x/2) 2k+ν k! Γ(ν + k + 1) , K ν (x)= π 2 I –ν (x)–I ν (x) sin(πν) , (see below for K ν (x) with ν = 0, 1, 2, ). 18.7.1-2. Some formulas. The modified Bessel functions possess the properties K –ν (x)=K ν (x); I –n (x)=(–1) n I n (x), n = 0, 1, 2, 2νI ν (x)=x[I ν–1 (x)–I ν+1 (x)], 2νK ν (x)=–x[K ν–1 (x)–K ν+1 (x)], d dx I ν (x)= 1 2 [I ν–1 (x)+I ν+1 (x)], d dx K ν (x)=– 1 2 [K ν–1 (x)+K ν+1 (x)]. 18.7.1-3. Modified Bessel functions for ν = n 1 2 ,wheren = 0, 1, 2, I 1/2 (x)= 2 πx sinh x, I –1/2 (x)= 2 πx cosh x, I 3/2 (x)= 2 πx – 1 x sinh x +coshx , I –3/2 (x)= 2 πx – 1 x cosh x +sinhx , I n+1/2 (x)= 1 √ 2πx e x n k=0 (–1) k (n + k)! k!(n – k)! (2x) k –(–1) n e –x n k=0 (n + k)! k!(n – k)! (2x) k , I –n–1/2 (x)= 1 √ 2πx e x n k=0 (–1) k (n + k)! k!(n – k)! (2x) k +(–1) n e –x n k=0 (n + k)! k!(n – k)! (2x) k , K 1/2 (x)= π 2x e –x , K 3/2 (x)= π 2x 1 + 1 x e –x , K n+1/2 (x)=K –n–1/2 (x)= π 2x e –x n k=0 (n + k)! k!(n – k)! (2x) k . 954 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.7.1-4. Modified Bessel functions for ν = n,wheren = 0, 1, 2, If ν = n is a nonnegative integer, then K n (x)=(–1) n+1 I n (x)ln x 2 + 1 2 n–1 m=0 (–1) m x 2 2m–n (n – m – 1)! m! + 1 2 (–1) n ∞ m=0 x 2 n+2m ψ(n + m + 1)+ψ(m + 1) m!(n + m)! ; n = 0, 1, 2, , where ψ(z) is the logarithmic derivative of the gamma function; for n = 0,thefirst sum is dropped. 18.7.1-5. Wronskians and similar formulas. W (I ν , I –ν )=– 2 πx sin(πν), W (I ν , K ν )=– 1 x , I ν (x)I –ν+1 (x)–I –ν (x)I ν–1 (x)=– 2 sin(πν) πx , I ν (x)K ν+1 (x)+I ν+1 (x)K ν (x)= 1 x , where W (f, g)=fg x – f x g. 18.7.2. Integral Representations and Asymptotic Expansions 18.7.2-1. Integral representations. The functions I ν (x)andK ν (x) can be represented in terms of definite integrals: I ν (x)= x ν π 1/2 2 ν Γ(ν + 1 2 ) 1 –1 exp(–xt)(1 – t 2 ) ν–1/2 dt (x > 0, ν >– 1 2 ), K ν (x)= ∞ 0 exp(–x cosh t)cosh(νt) dt (x > 0), K ν (x)= 1 cos 1 2 πν ∞ 0 cos(x sinh t)cosh(νt)dt (x > 0,–1 < ν < 1), K ν (x)= 1 sin 1 2 πν ∞ 0 sin(x sinh t)sinh(νt)dt (x > 0,–1 < ν < 1). For integer ν = n, I n (x)= 1 π π 0 exp(x cos t)cos(nt) dt (n = 0, 1, 2, ), K 0 (x)= ∞ 0 cos(x sinh t) dt = ∞ 0 cos(xt) √ t 2 + 1 dt (x > 0). . – 1. 18.6.3. Zeros and Orthogonality Properties of Bessel Functions 18.6.3-1. Zeros of Bessel functions. Each of the functions J ν (x)andY ν (x)hasinfinitely many real zeros (for real ν). All zeros are. Definitions. Basic Formulas 18.7.1-1. Modified Bessel functions of the first and the second kind. The modified Bessel functions of the first kind, I ν (x), and the modified Bessel functions of the second. Integral Representations and Asymptotic Expansions 18.6.2-1. Integral representations. The functions J ν (x)andY ν (x) can be represented in the form of definite integrals (for x > 0): πJ ν (x)= π 0 cos(x