Digital Library of Mathematical Functions
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NIST
10 Bessel FunctionsBessel and Hankel Functions

§10.12 Generating Function and Associated Series

For z and t\{0},

10.12.1 12z(t-t-1)=m=-tmJm(z).

Jacobi–Anger expansions: for z,θ,

10.12.2 cos(zsinθ) =J0(z)+2k=1J2k(z)cos(2kθ),
sin(zsinθ) =2k=0J2k+1(z)sin((2k+1)θ),
10.12.3 cos(zcosθ) =J0(z)+2k=1(-1)kJ2k(z)cos(2kθ),
sin(zcosθ) =2k=0(-1)kJ2k+1(z)cos((2k+1)θ).
10.12.4 1=J0(z)+2J2(z)+2J4(z)+2J6(z)+,
10.12.5 cosz =J0(z)-2J2(z)+2J4(z)-2J6(z)+,
sinz =2J1(z)-2J3(z)+2J5(z)-,
10.12.6 12zcosz =J1(z)-9J3(z)+25J5(z)-49J7(z)+,
12zsinz =4J2(z)-16J4(z)+36J6(z)-.