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10 Bessel FunctionsKelvin Functions

§10.71 Integrals

Contents

§10.71(i) Indefinite Integrals

In the following equations fν,gν is any one of the four ordered pairs given in (10.63.1), and f^ν,g^ν is either the same ordered pair or any other ordered pair in (10.63.1).

10.71.1 x1+νfνdx =-x1+ν2(fν+1-gν+1)
=-x1+ν(νxgν-gν),
10.71.2 x1-νfνdx =x1-ν2(fν-1-gν-1)
=x1-ν(νxgν+gν).
10.71.3 x(fνg^ν-gνf^ν)dx =x22(f^ν(fν+1+gν+1)-g^ν(fν+1-gν+1)-fν(f^ν+1+g^ν+1)+gν(f^ν+1-g^ν+1))
=12x(fνf^ν-fνf^ν+gνg^ν-gνg^ν),
10.71.4 x(fνg^ν+gνf^ν)dx =14x2(2fνg^ν-fν-1g^ν+1-fν+1g^ν-1+2gνf^ν-gν-1f^ν+1-gν+1f^ν-1).
10.71.5 x(fν2+gν2)dx=x(fνgν-fνgν)=-x2(fνfν+1+gνgν+1-fνgν+1+fν+1gν),
10.71.6 xfνgνdx =14x2(2fνgν-fν-1gν+1-fν+1gν-1),
10.71.7 x(fν2-gν2)dx =12x2(fν2-fν-1fν+1-gν2+gν-1gν+1).

Examples

10.71.8 xMν2(x)dx =x(berνxbeiνx-berνxbeiνx),
xNν2(x)dx =x(kerνxkeiνx-kerνxkeiνx),

where Mν(x) and Nν(x) are the modulus functions introduced in §10.68(i).

§10.71(ii) Definite Integrals

See Kerr (1978) and Glasser (1979).

§10.71(iii) Compendia

For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631).

For direct and inverse Laplace transforms of Kelvin functions see Prudnikov et al. (1992a, §3.19) and Prudnikov et al. (1992b, §3.19).