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10 Bessel FunctionsBessel and Hankel Functions

§10.9 Integral Representations


§10.9(i) Integrals along the Real Line

Bessel’s Integral

10.9.1 J0(z)=1π0πcos(zsinθ)θ=1π0πcos(zcosθ)θ,
10.9.2 Jn(z)=1π0πcos(zsinθ-nθ)θ=-nπ0πzcosθcos(nθ)θ,

Neumann’s Integral

10.9.3 Y0(z)=4π2012πcos(zcosθ)(γ+ln(2zsin2θ))θ,

where γ is Euler’s constant (§5.2(ii)).

Poisson’s and Related Integrals

10.9.4 Jν(z)=(12z)νπ12Γ(ν+12)0πcos(zcosθ)(sinθ)2νθ=2(12z)νπ12Γ(ν+12)01(1-t2)ν-12cos(zt)t,
10.9.5 Yν(z)=2(12z)νπ12Γ(ν+12)(01(1-t2)ν-12sin(zt)t-0-zt(1+t2)ν-12t),

Schläfli’s and Related Integrals

10.9.6 Jν(z)=1π0πcos(zsinθ-νθ)θ-sin(νπ)π0-zsinht-νtt,
10.9.7 Yν(z)=1π0πsin(zsinθ-νθ)θ-1π0(νt+-νtcos(νπ))-zsinhtt,

Mehler–Sonine and Related Integrals

10.9.8 Jν(x) =2π0sin(xcosht-12νπ)cosh(νt)t,
Yν(x) =-2π0cos(xcosht-12νπ)cosh(νt)t,

In particular,

10.9.9 J0(x) =2π0sin(xcosht)t,
Y0(x) =-2π0cos(xcosht)t,
10.9.10 Hν(1)(z)=-12νππ-zcosht-νtt,
10.9.11 Hν(2)(z)=-12νππ--zcosht-νtt,
10.9.12 Jν(x) =2(12x)-νπ12Γ(12-ν)1sin(xt)t(t2-1)ν+12,
Yν(x) =-2(12x)-νπ12Γ(12-ν)1cos(xt)t(t2-1)ν+12,
|ν|<12, x>0.
10.9.13 (z+ζz-ζ)12νJν((z2-ζ2)12)=1π0πζcosθcos(zsinθ-νθ)θ-sin(νπ)π0-ζcosht-zsinht-νtt,
10.9.14 (z+ζz-ζ)12νYν((z2-ζ2)12)=1π0πζcosθsin(zsinθ-νθ)θ-1π0(νt+ζcosht+-νt-ζcoshtcos(νπ))×-zsinhtt,
10.9.15 (z+ζz-ζ)12νHν(1)((z2-ζ2)12)=1π-12νπ-zcosht+ζsinht-νtt,
10.9.16 (z+ζz-ζ)12νHν(2)((z2-ζ2)12)=-1π12νπ--zcosht-ζsinht-νtt,

§10.9(ii) Contour Integrals

Schläfli–Sommerfeld Integrals

When |phz|<12π,

10.9.17 Jν(z)=12π-π+πzsinht-νtt,


10.9.18 Hν(1)(z) =1π-+πzsinht-νtt,
Hν(2)(z) =-1π--πzsinht-νtt.

Schläfli’s Integral

10.9.19 Jν(z)=(12z)ν2π-(0+)exp(t-z24t)ttν+1,

where the integration path is a simple loop contour, and tν+1 is continuous on the path and takes its principal value at the intersection with the positive real axis.

Hankel’s Integrals

In (10.9.20) and (10.9.21) the integration paths are simple loop contours not enclosing t=-1. Also, (t2-1)ν-12 is continuous on the path, and takes its principal value at the intersection with the interval (1,).

10.9.20 Jν(z)=Γ(12-ν)(12z)νπ320(1+)cos(zt)(t2-1)ν-12t,
10.9.21 Hν(1)(z) =Γ(12-ν)(12z)νπ321+(1+)zt(t2-1)ν-12t,
Hν(2)(z) =Γ(12-ν)(12z)νπ321-(1+)-zt(t2-1)ν-12t,

Mellin–Barnes Type Integrals

10.9.22 Jν(x)=12π-Γ(-t)(12x)ν+2tΓ(ν+t+1)t,
ν>0, x>0,

where the integration path passes to the left of t=0,1,2,.

10.9.23 Jν(z)=12π--c-+cΓ(t)Γ(ν-t+1)(12z)ν-2tt,

where c is a positive constant and the integration path encloses the points t=0,-1,-2,.

In (10.9.24) and (10.9.25) c is any constant exceeding max(ν,0).

10.9.24 Hν(1)(z) =--12νπ2π2c-c+Γ(t)Γ(t-ν)(-12z)ν-2tt,
10.9.25 Hν(2)(z) =12νπ2π2c-c+Γ(t)Γ(t-ν)(12z)ν-2tt,

For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116).

§10.9(iii) Products

10.9.26 Jμ(z)Jν(z)=2π0π/2Jμ+ν(2zcosθ)cos(μ-ν)θθ,
10.9.27 Jν(z)Jν(ζ)=2π0π/2J2ν(2(zζ)12sinθ)cos((z-ζ)cosθ)θ,

where the square root has its principal value.

10.9.28 Jν(z)Jν(ζ)=12πc-c+exp(12t-z2+ζ22t)Iν(zζt)tt,

where c is a positive constant. For the function Iν see §10.25(ii).

Mellin–Barnes Type

10.9.29 Jμ(x)Jν(x)=12π-Γ(-t)Γ(2t+μ+ν+1)(12x)μ+ν+2tΓ(t+μ+1)Γ(t+ν+1)Γ(t+μ+ν+1)t,

where the path of integration separates the poles of Γ(-t) from those of Γ(2t+μ+ν+1). See Paris and Kaminski (2001, p. 116) for related results.

Nicholson’s Integral

§10.9(iv) Compendia

For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).