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10 Bessel FunctionsBessel and Hankel Functions

§10.9 Integral Representations

Contents

§10.9(i) Integrals along the Real Line

Bessel’s Integral

10.9.1 J0(z)=1π0πcos(zsinθ)dθ=1π0πcos(zcosθ)dθ,
10.9.2 Jn(z)=1π0πcos(zsinθ-nθ)dθ=i-nπ0πeizcosθcos(nθ)dθ,
n.

Neumann’s Integral

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

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)2νdθ=2(12z)νπ12Γ(ν+12)01(1-t2)ν-12cos(zt)dt,
ν>-12.
10.9.5 Yν(z)=2(12z)νπ12Γ(ν+12)(01(1-t2)ν-12sin(zt)dt-0e-zt(1+t2)ν-12dt),
ν>-12,|phz|<12π.

Schläfli’s and Related Integrals

10.9.6 Jν(z)=1π0πcos(zsinθ-νθ)dθ-sin(νπ)π0e-zsinht-νtdt,
|phz|<12π,
10.9.7 Yν(z)=1π0πsin(zsinθ-νθ)dθ-1π0(eνt+e-νtcos(νπ))e-zsinhtdt,
|phz|<12π.

Mehler–Sonine and Related Integrals

10.9.8 Jν(x) =2π0sin(xcosht-12νπ)cosh(νt)dt,
Yν(x) =-2π0cos(xcosht-12νπ)cosh(νt)dt,
|ν|<1,x>0.

In particular,

10.9.9 J0(x) =2π0sin(xcosht)dt,
x>0,
Y0(x) =-2π0cos(xcosht)dt,
x>0.
10.9.10 Hν(1)(z)=e-12νπiπi-eizcosht-νtdt,
0<phz<π,
10.9.11 Hν(2)(z)=-e12νπiπi-e-izcosht-νtdt,
-π<phz<0.
10.9.12 Jν(x) =2(12x)-νπ12Γ(12-ν)1sin(xt)dt(t2-1)ν+12,
Yν(x) =-2(12x)-νπ12Γ(12-ν)1cos(xt)dt(t2-1)ν+12,
|ν|<12, x>0.
10.9.13 (z+ζz-ζ)12νJν((z2-ζ2)12)=1π0πeζcosθcos(zsinθ-νθ)dθ-sin(νπ)π0e-ζcosht-zsinht-νtdt,
(z+ζ)>0,
10.9.14 (z+ζz-ζ)12νYν((z2-ζ2)12)=1π0πeζcosθsin(zsinθ-νθ)dθ-1π0(eνt+ζcosht+e-νt-ζcoshtcos(νπ))×e-zsinhtdt,
(z±ζ)>0.
10.9.15 (z+ζz-ζ)12νHν(1)((z2-ζ2)12)=1πie-12νπi-eizcosht+iζsinht-νtdt,
(z±ζ)>0,
10.9.16 (z+ζz-ζ)12νHν(2)((z2-ζ2)12)=-1πie12νπi-e-izcosht-iζsinht-νtdt,
(z±ζ)<0.

§10.9(ii) Contour Integrals

Schläfli–Sommerfeld Integrals

When |phz|<12π,

10.9.17 Jν(z)=12πi-πi+πiezsinht-νtdt,

and

10.9.18 Hν(1)(z) =1πi-+πiezsinht-νtdt,
Hν(2)(z) =-1πi--πiezsinht-νtdt.

Schläfli’s Integral

10.9.19 Jν(z)=(12z)ν2πi-(0+)exp(t-z24t)dttν+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)νπ32i0(1+)cos(zt)(t2-1)ν-12dt,
ν12,32,.
10.9.21 Hν(1)(z) =Γ(12-ν)(12z)νπ32i1+i(1+)eizt(t2-1)ν-12dt,
Hν(2)(z) =Γ(12-ν)(12z)νπ32i1-i(1+)e-izt(t2-1)ν-12dt,
ν12,32,,|phz|<12π.

Mellin–Barnes Type Integrals

10.9.22 Jν(x)=12πi-iiΓ(-t)(12x)ν+2tΓ(ν+t+1)dt,
ν>0, x>0,

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

10.9.23 Jν(z)=12πi--ic-+icΓ(t)Γ(ν-t+1)(12z)ν-2tdt,

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) =-e-12νπi2π2c-ic+iΓ(t)Γ(t-ν)(-12iz)ν-2tdt,
0<phz<π,
10.9.25 Hν(2)(z) =e12νπi2π2c-ic+iΓ(t)Γ(t-ν)(12iz)ν-2tdt,
-π<phz<0.

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)θdθ,
(μ+ν)>-1.
10.9.27 Jν(z)Jν(ζ)=2π0π/2J2ν(2(zζ)12sinθ)cos((z-ζ)cosθ)dθ,
ν>-12,

where the square root has its principal value.

10.9.28 Jν(z)Jν(ζ)=12πic-ic+iexp(12t-z2+ζ22t)Iν(zζt)dtt,
ν>-1,

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

Mellin–Barnes Type

10.9.29 Jμ(x)Jν(x)=12πi-iiΓ(-t)Γ(2t+μ+ν+1)(12x)μ+ν+2tΓ(t+μ+1)Γ(t+ν+1)Γ(t+μ+ν+1)dt,
x>0,

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).