Digital Library of Mathematical Functions
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10 Bessel FunctionsModified Bessel Functions

§10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

10.32.1 I0(z)=1π0π±zcosθθ=1π0πcosh(zcosθ)θ.
10.32.2 Iν(z)=(12z)νπ12Γ(ν+12)0π±zcosθ(sinθ)2νθ=(12z)νπ12Γ(ν+12)-11(1-t2)ν-12±ztt,
ν>-12.
10.32.3 In(z)=1π0πzcosθcos(nθ)θ.
10.32.4 Iν(z)=1π0πzcosθcos(νθ)θ-sin(νπ)π0-zcosht-νtt,
|phz|<12π.
10.32.5 K0(z)=-1π0π±zcosθ(γ+ln(2z(sinθ)2))θ.
10.32.6 K0(x)=0cos(xsinht)t=0cos(xt)t2+1t,
x>0.
10.32.7 Kν(x)=sec(12νπ)0cos(xsinht)cosh(νt)t=csc(12νπ)0sin(xsinht)sinh(νt)t,
|ν|<1, x>0.
10.32.8 Kν(z)=π12(12z)νΓ(ν+12)0-zcosht(sinht)2νt=π12(12z)νΓ(ν+12)1-zt(t2-1)ν-12t,
ν>-12, |phz|<12π.
10.32.9 Kν(z)=0-zcoshtcosh(νt)t,
|phz|<12π.
10.32.10 Kν(z)=12(12z)ν0exp(-t-z24t)ttν+1,
|phz|<14π.

Basset’s Integral

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

§10.32(ii) Contour Integrals

10.32.12 Iν(z)=12π-π+πzcosht-νtt,
|phz|<12π.

Mellin–Barnes Type

10.32.13 Kν(z)=(12z)ν4πc-c+Γ(t)Γ(t-ν)(12z)-2tt,
c>max(ν,0),|phz|<π.
10.32.14 Kν(z)=12π2(π2z)12-zcos(νπ)-Γ(t)Γ(12-t-ν)Γ(12-t+ν)(2z)tt,
ν-12,|phz|<32π.

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

§10.32(iii) Products

10.32.15 Iμ(z)Iν(z)=2π012πIμ+ν(2zcosθ)cos((μ-ν)θ)θ,
(μ+ν)>-1.
10.32.16 Iμ(x)Kν(x)=0Jμ±ν(2xsinht)(-μ±ν)tt,
(μν)>-12, (μ±ν)>-1, x>0.
10.32.17 Kμ(z)Kν(z)=20Kμ±ν(2zcosht)cosh((μν)t)t,
|phz|<12π.
10.32.18 Kν(z)Kν(ζ)=120exp(-t2-z2+ζ22t)Kν(zζt)tt,
|phz|<π, |phζ|<π, |ph(z+ζ)|<14π.

Mellin–Barnes Type

10.32.19 Kμ(z)Kν(z)=18πc-c+Γ(t+12μ+12ν)Γ(t+12μ-12ν)Γ(t-12μ+12ν)Γ(t-12μ-12ν)Γ(2t)(12z)-2tt,
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).