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10 Bessel FunctionsModified Bessel Functions

§10.35 Generating Function and Associated Series

For z and t{0},

10.35.1 e12z(t+t1)=m=tmIm(z).

Jacobi–Anger expansions: for z,θ,

10.35.2 ezcosθ=I0(z)+2k=1Ik(z)cos(kθ),
10.35.3 ezsinθ=I0(z)+2k=0(1)kI2k+1(z)sin((2k+1)θ)+2k=1(1)kI2k(z)cos(2kθ).
10.35.4 1=I0(z)2I2(z)+2I4(z)2I6(z)+,
10.35.5 e±z=I0(z)±2I1(z)+2I2(z)±2I3(z)+,
10.35.6 coshz =I0(z)+2I2(z)+2I4(z)+2I6(z)+,
sinhz =2I1(z)+2I3(z)+2I5(z)+.