Handbook of mathematics for engineers and scienteists part 140 ppsx

18.3. SINE INTEGRAL AND COSINE INTEGRAL.FRESNEL INTEGRALS 941 18.2.3. Logarithmic Integral 18.2.3-1. Integral representations. Definition: li(x)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  x 0 dt ln t if 0 < x < 1, lim ε→+0   1–ε 0 dt ln t +  x 1+ε dt ln t  if x > 1. 18.2.3-2. Limiting properties. Relation to the exponential integral. For small x, li(x) ≈ x ln(1/x) . For large x, li(x) ≈ x ln x . Asymptotic expansion as x → 1: li(x)=C +ln|ln x| + ∞  k=1 ln k x k! k . Relation to the exponential integral: li x = Ei(ln x), x < 1; li(e x )=Ei(x), x < 0. 18.3. Sine Integral and Cosine Integral. Fresnel Integrals 18.3.1. Sine Integral 18.3.1-1. Integral representations. Properties. Definition: Si(x)=  x 0 sin t t dt,si(x)=–  ∞ x sin t t dt =Si(x)– π 2 . Specificvalues: Si(0)=0,Si(∞)= π 2 ,si(∞)=0. Properties: Si(–x)=–Si(x), si(x)+si(–x)=–π, lim x→–∞ si(x)=–π. 942 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.3.1-2. Expansions as x → 0 and x →∞. Expansion into series in powers of x as x → 0: Si(x)= ∞  k=1 (–1) k+1 x 2k–1 (2k – 1)(2k – 1)! . Asymptotic expansion as x →∞: si(x)=–cosx  M–1  m=0 (–1) m (2m)! x 2m+1 + O  |x| –2M–1   +sinx  N–1  m=1 (–1) m (2m – 1)! x 2m + O  |x| –2N   , where M, N = 1, 2, 18.3.2. Cosine Integral 18.3.2-1. Integral representations. Definition: Ci(x)=–  ∞ x cos t t dt = C +lnx +  x 0 cos t – 1 t dt, where C = 0.5772 is the Euler constant. 18.3.2-2. Expansions as x → 0 and x →∞. Expansion into series in powers of x as x → 0: Ci(x)=C +lnx + ∞  k=1 (–1) k x 2k 2k (2k)! . Asymptotic expansion as x →∞: Ci(x)=cosx  M–1  m=1 (–1) m (2m – 1)! x 2m +O  |x| –2M   +sinx  N–1  m=0 (–1) m (2m)! x 2m+1 +O  |x| –2N–1   , where M, N = 1, 2, 18.3.3. Fresnel Integrals 18.3.3-1. Integral representations. Definitions: S(x)= 1 √ 2π  x 0 sin t √ t dt =  2 π  √ x 0 sin t 2 dt, C(x)= 1 √ 2π  x 0 cos t √ t dt =  2 π  √ x 0 cos t 2 dt. 18.4. GAMMA FUNCTION,PSI FUNCTION, AND BETA FUNCTION 943 18.3.3-2. Expansions as x → 0 and x →∞. Expansion into series in powers of x as x → 0: S(x)=  2 π x ∞  k=0 (–1) k x 2k+1 (4k + 3)(2k + 1)! , C(x)=  2 π x ∞  k=0 (–1) k x 2k (4k + 1)(2k)! . Asymptotic expansion as x →∞: S(x)= 1 2 – cos x √ 2πx P (x)– sin x √ 2πx Q(x), C(x)= 1 2 + sin x √ 2πx P (x)– cos x √ 2πx Q(x), P (x)=1 – 1×3 (2x) 2 + 1×3×5×7 (2x) 4 – ···, Q(x)= 1 2x – 1×3×5 (2x) 3 + ··· . 18.4. Gamma Function, Psi Function, and Beta Function 18.4.1. Gamma Function 18.4.1-1. Integral representations. Simplest properties. The gamma function, Γ(z), is an analytic function of the complex argument z everywhere except for the points z = 0,–1,–2, For Re z > 0, Γ(z)=  ∞ 0 t z–1 e –t dt. For –(n + 1)<Rez <–n,wheren = 0,1,2, , Γ(z)=  ∞ 0  e –t – n  m=0 (–1) m m!  t z–1 dt. Simplest properties: Γ(z + 1)=zΓ(z), Γ(n + 1)=n!, Γ(1)=Γ(2)=1. Fractional values of the argument: Γ  1 2  = √ π, Γ  – 1 2  =–2 √ π, Γ  n + 1 2  = √ π 2 n (2n – 1)!!, Γ  1 2 – n  =(–1) n 2 n √ π (2n – 1)!! . 944 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.4.1-2. Euler, Stirling, and other formulas. Euler formula Γ(z) = lim n→∞ n! n z z(z + 1) (z + n) (z ≠ 0,–1,–2, ). Symmetry formulas: Γ(z)Γ(–z)=– π z sin(πz) , Γ(z)Γ(1 – z)= π sin(πz) , Γ  1 2 + z  Γ  1 2 – z  = π cos(πz) . Multiple argument formulas: Γ(2z)= 2 2z–1 √ π Γ(z)Γ  z + 1 2  , Γ(3z)= 3 3z–1/2 2π Γ(z)Γ  z + 1 3  Γ  z + 2 3  , Γ(nz)=(2π) (1–n)/2 n nz–1/2 n–1  k=0 Γ  z + k n  . Asymptotic expansion (Stirling formula): Γ(z)= √ 2πe –z z z–1/2  1 + 1 12 z –1 + 1 288 z –2 + O(z –3 )  (|arg z| < π). 18.4.2. Psi Function (Digamma Function) 18.4.2-1. Definition. Integral representations. Definition: ψ(z)= d ln Γ(z) dz = Γ  z (z) Γ(z) . The psi function is the logarithmic derivative of the gamma function and is also called the digamma function. Integral representations (Re z > 0): ψ(z)=  ∞ 0  e –t –(1 + t) –z  t –1 dt, ψ(z)=lnz +  ∞ 0  t –1 –(1 – e –t ) –1  e –tz dt, ψ(z)=–C +  1 0 1 – t z–1 1 – t dt, where C =–ψ(1)=0.5772 is the Euler constant. Values for integer argument: ψ(1)=–C, ψ(n)=–C + n–1  k=1 k –1 (n = 2, 3, ). 18.4. GAMMA FUNCTION,PSI FUNCTION, AND BETA FUNCTION 945 18.4.2-2. Properties. Asymptotic expansion as z →∞. Functional relations: ψ(z)–ψ(1 + z)=– 1 z , ψ(z)–ψ(1 – z)=–π cot(πz), ψ(z)–ψ(–z)=–π cot(πz)– 1 z , ψ  1 2 + z  – ψ  1 2 – z  = π tan(πz), ψ(mz)=lnm + 1 m m–1  k=0 ψ  z + k m  . Asymptotic expansion as z →∞(|arg z| < π): ψ(z)=lnz – 1 2z – 1 12z 2 + 1 120z 4 – 1 252z 6 + ···=lnz – 1 2z – ∞  n=1 B 2n 2nz 2n , where the B 2n are Bernoulli numbers. 18.4.3. Beta Function 18.4.3-1. Integral representation. Relationship with the gamma function. Definition: B(x, y)=  1 0 t x–1 (1 – t) y–1 dt, where Re x > 0 and Re y > 0. Relationship with the gamma function: B(x, y)= Γ(x)Γ(y) Γ(x + y) . 18.4.3-2. Some properties. B(x, y)=B(y, x); B(x, y + 1)= y x B(x + 1, y)= y x + y B(x, y); B(x, 1 – x)= π sin(πx) , 0 < x < 1; 1 B(n, m) = mC n–1 n+m–1 = nC m–1 n+m–1 , where n and m are positive integers. 946 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.5. Incomplete Gamma and Beta Functions 18.5.1. Incomplete Gamma Function 18.5.1-1. Integral representations. Recurrence formulas. Definitions: γ(α, x)=  x 0 e –t t α–1 dt,Reα > 0, Γ(α, x)=  ∞ x e –t t α–1 dt = Γ(α)–γ(α, x). Recurrence formulas: γ(α + 1, x)=αγ(α, x)–x α e –x , γ(α + 1, x)=(x + α)γ(α, x)+(1 – α)xγ(α – 1, x), Γ(α + 1, x)=αΓ(α, x)+x α e –x . Special cases: γ(n + 1, x)=n!  1 – e –x  n  k=0 x k k!  , n = 0, 1, ; Γ(n + 1, x)=n! e –x n  k=0 x k k! , n = 0, 1, ; Γ(–n, x)= (–1) n n!  Γ(0, x)–e –x n–1  k=0 (–1) k k! x k+1  , n = 1, 2, 18.5.1-2. Expansions as x → 0 and x →∞. Relation to other functions. Asymptotic expansions as x → 0: γ(α, x)= ∞  n=0 (–1) n x α+n n!(α + n) , Γ(α, x)=Γ(α)– ∞  n=0 (–1) n x α+n n!(α + n) . Asymptotic expansions as x →∞: γ(α, x)=Γ(α)–x α–1 e –x  M–1  m=0 (1 – α) m (–x) m + O  |x| –M   , Γ(α, x)=x α–1 e –x  M–1  m=0 (1 – α) m (–x) m + O  |x| –M    – 3 2 π <argx < 3 2 π  . 18.6. BESSEL FUNCTIONS (CYLINDRICAL FUNCTIONS) 947 Asymptotic formulas as α →∞: γ(x, α)=Γ(α)  Φ  2 √ x – √ α – 1  + O  1 √ α  , Φ(x)= 1 √ 2π  x –∞ exp  – 1 2 t 2  dt; γ(x, α)=Γ(α)  Φ  3 √ αz  + O  1 α  , z =  x α  1/3 – 1 + 1 9α . Representation of the error function, complementary error function, and exponential integral in terms of the gamma functions: erf x = 1 √ π γ  1 2 , x 2  , erfc x = 1 √ π Γ  1 2 , x 2  ,Ei(–x)=–Γ(0, x). 18.5.2. Incomplete Beta Function 18.5.2-1. Integral representation. Definitions: B x (a, b)=  x 0 t a–1 (1 – t) b–1 dt, I x (a, b)= B x (a, b) B(a, b) , where Re a > 0 and Re b > 0,andB(a, b)=B 1 (a, b) is the beta function. 18.5.2-2. Some properties. Symmetry: I x (a, b)+I 1–x (b, a)=1. recurrence formulas: I x (a, b)=xI x (a – 1, b)+(1 – x)I x (a, b – 1), (a + b)I x (a, b)=aI x (a + 1, b)+bI x (a, b + 1), (a + b – ax)I x (a, b)=a(1 – x)I x (a + 1, b – 1)+bI x (a, b + 1). 18.6. Bessel Functions (Cylindrical Functions) 18.6.1. Definitions and Basic Formulas 18.6.1-1. Bessel functions of the first and the second kind. The Bessel function of the first kind, J ν (x), and the Bessel function of the second kind, Y ν (x) (also called the Neumann function), are solutions of the Bessel equation x 2 y  xx + xy  x +(x 2 – ν 2 )y = 0 and are defined by the formulas J ν (x)= ∞  k=0 (–1) k (x/2) ν+2k k! Γ(ν + k + 1) , Y ν (x)= J ν (x)cosπν – J –ν (x) sin πν .(18.6.1.1) The formula for Y ν (x) is valid for ν ≠ 0, 1, 2, (the cases ν ≠ 0, 1, 2, are discussed in what follows). The general solution of the Bessel equation has the form Z ν (x)=C 1 J ν (x)+C 2 Y ν (x) and is called the cylinder function. . Functions) 18.6.1. Definitions and Basic Formulas 18.6.1-1. Bessel functions of the first and the second kind. The Bessel function of the first kind, J ν (x), and the Bessel function of the second kind, Y ν (x). πν .(18.6.1.1) The formula for Y ν (x) is valid for ν ≠ 0, 1, 2, (the cases ν ≠ 0, 1, 2, are discussed in what follows). The general solution of the Bessel equation has the form Z ν (x)=C 1 J ν (x)+C 2 Y ν (x) and. AND THEIR PROPERTIES 18.4.1-2. Euler, Stirling, and other formulas. Euler formula Γ(z) = lim n→∞ n! n z z(z + 1) (z + n) (z ≠ 0,–1,–2, ). Symmetry formulas: Γ(z)Γ(–z)=– π z sin(πz) , Γ(z)Γ(1 –