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10 Bessel FunctionsSpherical Bessel Functions

§10.54 Integral Representations

10.54.1 jn(z)=zn2n+1n!0πcos(zcosθ)(sinθ)2n+1dθ.
10.54.2 jn(z) =(-i)n20πeizcosθPn(cosθ)sinθdθ.
10.54.3 kn(z) =π21e-ztPn(t)dt,
|phz|<12π.
10.54.4 jn(z) =(-i)n+12πi(-1+,1+)eiztQn(t)dt,
|phz|<12π.
10.54.5 hn(1)(z) =(-i)n+1πi(1+)eiztQn(t)dt,
hn(2)(z) =(-i)n+1πi(-1+)eiztQn(t)dt,
|phz|<12π.

For the Legendre polynomial Pn and the associated Legendre function Qn see §§18.3 and 14.21(i), with μ=0 and ν=n.

Additional integral representations can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.9 and §10.32.