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11 Struve and Related FunctionsRelated Functions

§11.10 Anger–Weber Functions

  1. §11.10(i) Definitions
  2. §11.10(ii) Differential Equations
  3. §11.10(iii) Maclaurin Series
  4. §11.10(iv) Graphics
  5. §11.10(v) Interrelations
  6. §11.10(vi) Relations to Other Functions
  7. §11.10(vii) Special Values
  8. §11.10(viii) Expansions in Series of Products of Bessel Functions
  9. §11.10(ix) Recurrence Relations and Derivatives
  10. §11.10(x) Integrals and Sums

§11.10(i) Definitions

The Anger function 𝐉ν(z) and Weber function 𝐄ν(z) are defined by

11.10.1 𝐉ν(z)=1π0πcos(νθzsinθ)dθ,
11.10.2 𝐄ν(z)=1π0πsin(νθzsinθ)dθ.

Each is an entire function of z and ν. Also,

11.10.3 1π02πcos(νθzsinθ)dθ=(1+cos(2πν))𝐉ν(z)+sin(2πν)𝐄ν(z).

The associated Anger–Weber function 𝐀ν(z) is defined by

11.10.4 𝐀ν(z)=1π0eνtzsinhtdt,

(11.10.4) also applies when z=0 and ν>0.

§11.10(ii) Differential Equations

The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation

11.10.5 d2wdz2+1zdwdz+(1ν2z2)w=f(ν,z),


11.10.6 f(ν,z)=(zν)πz2sin(πν),


11.10.7 f(ν,z)=1πz2(z+ν+(zν)cos(πν)),

§11.10(iii) Maclaurin Series

11.10.8 𝐉ν(z)=cos(12πν)S1(ν,z)+sin(12πν)S2(ν,z),
11.10.9 𝐄ν(z)=sin(12πν)S1(ν,z)cos(12πν)S2(ν,z),


11.10.10 S1(ν,z)=k=0(1)k(12z)2kΓ(k+12ν+1)Γ(k12ν+1),
11.10.11 S2(ν,z)=k=0(1)k(12z)2k+1Γ(k+12ν+32)Γ(k12ν+32).

These expansions converge absolutely for all finite values of z.

§11.10(iv) Graphics

See accompanying text
Figure 11.10.1: Anger function 𝐉ν(x) for 8x8 and ν=0,12,1,32. Magnify
See accompanying text
Figure 11.10.2: Weber function 𝐄ν(x) for 8x8 and ν=0,12,1,32. Magnify
See accompanying text
Figure 11.10.3: Anger function 𝐉ν(x) for 10x10 and 0ν5. Magnify 3D Help
See accompanying text
Figure 11.10.4: Weber function 𝐄ν(x) for 10x10 and 0ν5. Magnify 3D Help

§11.10(v) Interrelations

11.10.12 𝐉ν(z) =𝐉ν(z),
𝐄ν(z) =𝐄ν(z).
11.10.13 sin(πν)𝐉ν(z) =cos(πν)𝐄ν(z)𝐄ν(z),
11.10.14 sin(πν)𝐄ν(z) =𝐉ν(z)cos(πν)𝐉ν(z).
11.10.15 𝐉ν(z)=Jν(z)+sin(πν)𝐀ν(z),
11.10.16 𝐄ν(z)=Yν(z)cos(πν)𝐀ν(z)𝐀ν(z).

§11.10(vi) Relations to Other Functions

11.10.17 𝐉ν(z) =sin(πν)π(s0,ν(z)νs1,ν(z)),
11.10.18 𝐄ν(z) =1π(1+cos(πν))s0,ν(z)νπ(1cos(πν))s1,ν(z).
11.10.19 𝐉12(z) =𝐄12(z)=(12πz)12(A+(χ)coszA(χ)sinz),
11.10.20 𝐉12(z) =𝐄12(z)=(12πz)12(A+(χ)sinz+A(χ)cosz),


11.10.21 A±(χ) =C(χ)±S(χ),
χ =(2z/π)12.

For the Fresnel integrals C and S see §7.2(iii).

For n=1,2,3,,

11.10.22 𝐄n(z)=𝐇n(z)+1πk=0m1Γ(k+12)Γ(n+12k)(12z)n2k1,


11.10.23 𝐄n(z)=𝐇n(z)+(1)n+1πk=0m2Γ(nk12)Γ(k+32)(12z)n+2k+1,


11.10.24 m1 =12n12,
m2 =12n32.

§11.10(vii) Special Values

11.10.25 𝐉ν(0) =sin(πν)πν, 𝐄ν(0) =1cos(πν)πν.
11.10.26 𝐄0(z) =𝐇0(z), 𝐄1(z) =2π𝐇1(z).
11.10.29 𝐉n(z)=Jn(z),

§11.10(viii) Expansions in Series of Products of Bessel Functions

11.10.30 𝐉ν(z)=2sin(12νπ)k=0(1)kJk12ν+12(12z)Jk+12ν+12(12z)+2cos(12νπ)k=0(1)kJk12ν(12z)Jk+12ν(12z),
11.10.31 𝐄ν(z)=2cos(12νπ)k=0(1)kJk12ν+12(12z)Jk+12ν+12(12z)+2sin(12νπ)k=0(1)kJk12ν(12z)Jk+12ν(12z),

where the prime on the second summation symbols means that the first term is to be halved.

§11.10(ix) Recurrence Relations and Derivatives

11.10.32 𝐉ν1(z)+𝐉ν+1(z)=2νz𝐉ν(z)2πzsin(πν),
11.10.33 𝐄ν1(z)+𝐄ν+1(z)=2νz𝐄ν(z)2πz(1cos(πν)).
11.10.34 2𝐉ν(z) =𝐉ν1(z)𝐉ν+1(z),
11.10.35 2𝐄ν(z) =𝐄ν1(z)𝐄ν+1(z),
11.10.36 z𝐉ν(z)±ν𝐉ν(z)=±z𝐉ν1(z)±sin(πν)π,
11.10.37 z𝐄ν(z)±ν𝐄ν(z)=±z𝐄ν1(z)±(1cos(πν))π.

§11.10(x) Integrals and Sums

For collections of integral representations and integrals see Erdélyi et al. (1954a, §§4.19 and 5.17), Marichev (1983, pp. 194–195 and 214–215), Oberhettinger (1972, p. 128), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105 and 189–190), Prudnikov et al. (1990, §§1.5 and 2.8), Prudnikov et al. (1992a, §3.18), Prudnikov et al. (1992b, §3.18), and Zanovello (1977).

For sums see Hansen (1975, pp. 456–457) and Prudnikov et al. (1990, §§6.4.2–6.4.3).