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10 Bessel FunctionsKelvin Functions

§10.71 Integrals

  1. §10.71(i) Indefinite Integrals
  2. §10.71(ii) Definite Integrals
  3. §10.71(iii) Compendia

§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+1gν+1)=x1+ν(νxgνgν),
10.71.2 x1νfνdx =x1ν2(fν1gν1)=x1ν(νxgν+gν).
10.71.3 x(fνg^νgνf^ν)dx =x22(f^ν(fν+1+gν+1)g^ν(fν+1gν+1)fν(f^ν+1+g^ν+1)+gν(f^ν+1g^ν+1))=12x(fνf^νfνf^ν+gνg^νgνg^ν),
10.71.4 x(fνg^ν+gνf^ν)dx =14x2(2fνg^νfν1g^ν+1fν+1g^ν1+2gνf^νgν1f^ν+1gν+1f^ν1).
10.71.5 x(fν2+gν2)dx=x(fνgνfνgν)=x2(fνfν+1+gνgν+1fνgν+1+fν+1gν),
10.71.6 xfνgνdx =14x2(2fνgνfν1gν+1fν+1gν1),
10.71.7 x(fν2gν2)dx =12x2(fν2fν1fν+1gν2+gν1gν+1).


10.71.8 xMν2(x)dx =x(berνxbeiνxberνxbeiνx),
xNν2(x)dx =x(kerνxkeiνxkerνxkeiνx),

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).