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10 Bessel FunctionsModified Bessel Functions

§10.32 Integral Representations

Contents

§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(1-t2)ν-12e±ztdt,
ν>-12.
10.32.3 In(z)=1π0πezcosθcos(nθ)dθ.
10.32.4 Iν(z)=1π0πezcosθcos(νθ)dθ-sin(νπ)π0e-zcosht-νtdt,
|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)0e-zcosht(sinht)2νdt=π12(12z)νΓ(ν+12)1e-zt(t2-1)ν-12dt,
ν>-12, |phz|<12π.
10.32.9 Kν(z)=0e-zcoshtcosh(νt)dt,
|phz|<12π.
10.32.10 Kν(z)=12(12z)ν0exp(-t-z24t)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

10.32.12 Iν(z)=12πi-iπ+iπezcosht-νtdt,
|phz|<12π.

Mellin–Barnes Type

10.32.13 Kν(z)=(12z)ν4πic-ic+iΓ(t)Γ(t-ν)(12z)-2tdt,
c>max(ν,0),|phz|<π.
10.32.14 Kν(z)=12π2i(π2z)12e-zcos(νπ)-iiΓ(t)Γ(12-t-ν)Γ(12-t+ν)(2z)tdt,
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)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(-t2-z2+ζ22t)Kν(zζt)dtt,
|phz|<π, |phζ|<π, |ph(z+ζ)|<14π.

Mellin–Barnes Type

10.32.19 Kμ(z)Kν(z)=18πic-ic+iΓ(t+12μ+12ν)Γ(t+12μ-12ν)Γ(t-12μ+12ν)Γ(t-12μ-12ν)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).