Digital Library of Mathematical Functions
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NIST
10 Bessel FunctionsModified Bessel Functions

§10.35 Generating Function and Associated Series

For z and t\{0},

10.35.1 12z(t+t-1)=m=-tmIm(z).

Jacobi–Anger expansions: for z,θ,

10.35.2 zcosθ=I0(z)+2k=1Ik(z)cos(kθ),
10.35.3 zsinθ=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 ±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)+.