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10 Bessel FunctionsModified Bessel Functions

§10.32 Integral Representations

Contents
  1. §10.32(i) Integrals along the Real Line
  2. §10.32(ii) Contour Integrals
  3. §10.32(iii) Products
  4. §10.32(iv) Compendia

§10.32(i) Integrals along the Real Line

10.32.1 I0(z)=1π0πe±zcosθdθ=1π0πcosh(zcosθ)dθ.
10.32.2 Iν(z)=(12z)νπ12Γ(ν+12)0πe±zcosθ(sinθ)2νdθ=(12z)νπ12Γ(ν+12)11(1t2)ν12e±ztdt,
ν>12.
10.32.3 In(z)=1π0πezcosθcos(nθ)dθ.
10.32.4 Iν(z)=1π0πezcosθcos(νθ)dθsin(νπ)π0ezcoshtνtdt,
|phz|<12π.
10.32.5 K0(z)=1π0πe±zcosθ(γ+ln(2z(sinθ)2))dθ.
10.32.6 K0(x)=0cos(xsinht)dt=0cos(xt)t2+1dt,
x>0.
10.32.7 Kν(x)=sec(12νπ)0cos(xsinht)cosh(νt)dt=csc(12νπ)0sin(xsinht)sinh(νt)dt,
|ν|<1, x>0.
10.32.8 Kν(z)=π12(12z)νΓ(ν+12)0ezcosht(sinht)2νdt=π12(12z)νΓ(ν+12)1ezt(t21)ν12dt,
ν>12, |phz|<12π.
10.32.9 Kν(z)=0ezcoshtcosh(νt)dt,
|phz|<12π.
10.32.10 Kν(z)=12(12z)ν0exp(tz24t)dttν+1,
|phz|<14π.

Basset’s Integral

10.32.11 Kν(xz)=Γ(ν+12)(2z)νπ12xν0cos(xt)dt(t2+z2)ν+12,
ν>12, x>0, |phz|<12π.

§10.32(ii) Contour Integrals

Mellin–Barnes Type

10.32.13 Kν(z)=(12z)ν4πicic+iΓ(t)Γ(tν)(12z)2tdt,
c>max(ν,0),|phz|<12π.
10.32.14 Kν(z)=12π2i(π2z)12ezcos(νπ)iiΓ(t)Γ(12tν)Γ(12t+ν)(2z)tdt,
ν12,|phz|<32π.

In (10.32.14) the integration contour separates the poles of Γ(t) from the poles of Γ(12tν)Γ(12t+ν).

§10.32(iii) Products

10.32.15 Iμ(z)Iν(z)=2π012πIμ+ν(2zcosθ)cos((μν)θ)dθ,
(μ+ν)>1.
10.32.16 Iμ(x)Kν(x)=0Jμ±ν(2xsinht)e(μ±ν)tdt,
(μν)>12, (μ±ν)>1, x>0.
10.32.17 Kμ(z)Kν(z)=20Kμ±ν(2zcosht)cosh((μν)t)dt,
|phz|<12π.
10.32.18 Kν(z)Kν(ζ)=120exp(t2z2+ζ22t)Kν(zζt)dtt,
|phz|<π, |phζ|<π, |ph(z+ζ)|<14π.

Mellin–Barnes Type

10.32.19 Kμ(z)Kν(z)=18πicic+iΓ(t+12μ+12ν)Γ(t+12μ12ν)Γ(t12μ+12ν)Γ(t12μ12ν)Γ(2t)(12z)2tdt,
c>12(|μ|+|ν|),|phz|<12π.

For similar integrals for Jν(z)Kν(z) and Iν(z)Kν(z) see Paris and Kaminski (2001, p. 116).

§10.32(iv) Compendia

For collections of integral representations of modified Bessel functions, or products of modified Bessel functions, see Erdélyi et al. (1953b, §§7.3, 7.12, and 7.14.2), Erdélyi et al. (1954a, pp. 48–60, 105–115, 276–285, and 357–359), Gröbner and Hofreiter (1950, pp. 193–194), Magnus et al. (1966, §3.7), Marichev (1983, pp. 191–216), and Watson (1944, Chapters 6, 12, and 13).