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11 Struve and Related FunctionsRelated Functions

§11.10 Anger–Weber Functions

Contents

§11.10(i) Definitions

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

11.10.1 Jν(z)=1π0πcos(νθ-zsinθ)dθ,
11.10.2 Eν(z)=1π0πsin(νθ-zsinθ)dθ.

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

11.10.3 1π02πcos(νθ-zsinθ)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)=1π0e-νt-zsinhtdt,
z>0.

(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+(1-ν2z2)w=f(ν,z),

where

11.10.6 f(ν,z)=(z-ν)πz2sin(πν),
w=Jν(z),

or

11.10.7 f(ν,z)=-1πz2(z+ν+(z-ν)cos(πν)),
w=Eν(z).

§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),

where

11.10.10 S1(ν,z)=k=0(-1)k(12z)2kΓ(k+12ν+1)Γ(k-12ν+1),
11.10.11 S2(ν,z)=k=0(-1)k(12z)2k+1Γ(k+12ν+32)Γ(k-12ν+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
Figure 11.10.3: Anger function Jν(x) for -10x10 and 0ν5. Magnify
Figure 11.10.4: Weber function Eν(x) for -10x10 and 0ν5. Magnify

§11.10(v) Interrelations

11.10.12 Jν(-z) =J-ν(z),
Eν(-z) =-E-ν(z).
11.10.13 sin(πν)Jν(z) =cos(πν)Eν(z)-E-ν(z),
11.10.14 sin(πν)Eν(z) =J-ν(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)-A-ν(z).

§11.10(vi) Relations to Other Functions

11.10.17 Jν(z) =sin(πν)π(s0,ν(z)-νs-1,ν(z)),
11.10.18 Eν(z) =-1π(1+cos(πν))s0,ν(z)-νπ(1-cos(πν))s-1,ν(z).
11.10.19 J-12(z) =E12(z)
=(12πz)-12(A+(χ)cosz-A-(χ)sinz),
11.10.20 J12(z) =-E-12(z)
=(12πz)-12(A+(χ)sinz+A-(χ)cosz),

where

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+12-k)(12z)n-2k-1,

and

11.10.23 E-n(z)=-H-n(z)+(-1)n+1πk=0m2Γ(n-k-12)Γ(k+32)(12z)-n+2k+1,

where

11.10.24 m1 =12n-12,
m2 =12n-32.

§11.10(vii) Special Values

11.10.25 Jν(0) =sin(πν)πν, Eν(0) =1-cos(πν)πν.
11.10.26 E0(z) =-H0(z), E1(z) =2π-H1(z).

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

11.10.30 Jν(z)=2sin(12νπ)k=0(-1)kJk-12ν+12(12z)Jk+12ν+12(12z)+2cos(12νπ)k=0(-1)kJk-12ν(12z)Jk+12ν(12z),
11.10.31 Eν(z)=-2cos(12νπ)k=0(-1)kJk-12ν+12(12z)Jk+12ν+12(12z)+2sin(12νπ)k=0(-1)kJk-12ν(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 Jν-1(z)+Jν+1(z)=2νzJν(z)-2πzsin(πν),
11.10.33 Eν-1(z)+Eν+1(z)=2νzEν(z)-2πz(1-cos(πν)).
11.10.34 2Jν(z) =Jν-1(z)-Jν+1(z),
11.10.35 2Eν(z) =Eν-1(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)±(1-cos(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).