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10 Bessel FunctionsBessel and Hankel Functions

§10.13 Other Differential Equations

In the following equations ν,λ,p,q, and r are real or complex constants with λ0, p0, and q0.

10.13.1 w′′+(λ2-ν2-14z2)w=0,
w=z12𝒞ν(λz),
10.13.2 w′′+(λ24z-ν2-14z2)w=0,
w=z12𝒞ν(λz12),
10.13.3 w′′+λ2zp-2w=0,
w=z12𝒞1/p(2λz12p/p),
10.13.4 w′′+12νzw+λ2w=0,
w=z±ν𝒞ν(λz),
10.13.5 z2w′′+(1-2r)zw+(λ2q2z2q+r2-ν2q2)w=0,
w=zr𝒞ν(λzq),
10.13.6 w′′+(λ2e2z-ν2)w=0,
w=𝒞ν(λez),
10.13.7 z2(z2-ν2)w′′+z(z2-3ν2)w+((z2-ν2)2-(z2+ν2))w=0,
w=𝒞ν(z),
10.13.8 w(2n)=(-1)nλ2nz-nw,
w=z12n𝒞n(2λekπi/nz12), k=0,1,,2n-1.

In (10.13.9)–(10.13.11) 𝒞ν(z), 𝒟μ(z) are any cylinder functions of orders ν,μ, respectively, and ϑ=z(d/dz).

10.13.9 z2w′′′+3zw′′+(4z2+1-4ν2)w+4zw=0,
w=𝒞ν(z)𝒟ν(z),
10.13.10 z3w′′′+z(4z2+1-4ν2)w+(4ν2-1)w=0,
w=z𝒞ν(z)𝒟ν(z),
10.13.11 (ϑ4-2(ν2+μ2)ϑ2+(ν2-μ2)2)w+4z2(ϑ+1)(ϑ+2)w=0,
w=𝒞ν(z)𝒟μ(z).

For further differential equations see Kamke (1977, pp. 440–451). See also Watson (1944, pp. 95–100).