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10 Bessel FunctionsBessel and Hankel Functions

§10.12 Generating Function and Associated Series

For z and t{0},

10.12.1 e12z(tt1)=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).