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10 Bessel FunctionsBessel and Hankel Functions

§10.9 Integral Representations

  1. §10.9(i) Integrals along the Real Line
  2. §10.9(ii) Contour Integrals
  3. §10.9(iii) Products
  4. §10.9(iv) Compendia

§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(zsinθnθ)dθ=inπ0πeizcosθcos(nθ)dθ,

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(1t2)ν12cos(zt)dt,
10.9.5 Yν(z)=2(12z)νπ12Γ(ν+12)(01(1t2)ν12sin(zt)dt0ezt(1+t2)ν12dt),

Schläfli’s and Related Integrals

10.9.6 Jν(z)=1π0πcos(zsinθνθ)dθsin(νπ)π0ezsinhtνtdt,
10.9.7 Yν(z)=1π0πsin(zsinθνθ)dθ1π0(eνt+eνtcos(νπ))ezsinhtdt,

Mehler–Sonine and Related Integrals

10.9.8 Jν(x) =2π0sin(xcosht12νπ)cosh(νt)dt,
Yν(x) =2π0cos(xcosht12νπ)cosh(νt)dt,

In particular,

10.9.9 J0(x) =2π0sin(xcosht)dt,
Y0(x) =2π0cos(xcosht)dt,
10.9.10 Hν(1)(z)=e12νπiπieizcoshtνtdt,
10.9.11 Hν(2)(z)=e12νπiπieizcoshtνtdt,
10.9.12 Jν(x) =2(12x)νπ12Γ(12ν)1sin(xt)dt(t21)ν+12,
Yν(x) =2(12x)νπ12Γ(12ν)1cos(xt)dt(t21)ν+12,
|ν|<12, x>0.
10.9.13 (z+ζzζ)12νJν((z2ζ2)12)=1π0πeζcosθcos(zsinθνθ)dθsin(νπ)π0eζcoshtzsinhtνtdt,
10.9.14 (z+ζzζ)12νYν((z2ζ2)12)=1π0πeζcosθsin(zsinθνθ)dθ1π0(eνt+ζcosht+eνtζcoshtcos(νπ))ezsinhtdt,
10.9.15 (z+ζzζ)12νHν(1)((z2ζ2)12)=1πie12νπieizcosht+iζsinhtνtdt,
10.9.16 (z+ζzζ)12νHν(2)((z2ζ2)12)=1πie12νπieizcoshtiζsinhtνtdt,

§10.9(ii) Contour Integrals

Schläfli–Sommerfeld Integrals

Schläfli’s Integral

10.9.19 Jν(z)=(12z)ν2πi(0+)exp(tz24t)dttν+1,

where the integration path is a simple loop contour (see Figure 5.9.1), 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, (t21)ν12 is continuous on the path, and takes its principal value at the intersection with the interval (1,).

10.9.20 Jν(z)=Γ(12ν)(12z)νπ32i0(1+)cos(zt)(t21)ν12dt,
10.9.21 Hν(1)(z) =Γ(12ν)(12z)νπ32i1+i(1+)eizt(t21)ν12dt,
Hν(2)(z) =Γ(12ν)(12z)νπ32i1i(1+)eizt(t21)ν12dt,

Mellin–Barnes Type Integrals

10.9.22 Jν(x)=12πiiiΓ(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)=12πiic+icΓ(t)Γ(νt+1)(12z)ν2tdt,

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) =e12νπi2π2cic+iΓ(t)Γ(tν)(12iz)ν2tdt,
10.9.25 Hν(2)(z) =e12νπi2π2cic+iΓ(t)Γ(tν)(12iz)ν2tdt,

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((μν)θ)dθ,
10.9.27 Jν(z)Jν(ζ)=2π0π/2J2ν(2(zζ)12sinθ)cos((zζ)cosθ)dθ,

where the square root has its principal value.

10.9.28 Jν(z)Jν(ζ)=12πicic+iexp(12tz2+ζ22t)Iν(zζt)dtt,

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

Mellin–Barnes Type

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

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).