Digital Library of Mathematical Functions
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10 Bessel FunctionsModified Bessel Functions

§10.31 Power Series

For Iν(z) see (10.25.2) and (10.27.1). When ν is not an integer the corresponding expansion for Kν(z) is obtained from (10.25.2) and (10.27.4).

When n=0,1,2,,

10.31.1 Kn(z)=12(12z)-nk=0n-1(n-k-1)!k!(-14z2)k+(-1)n+1ln(12z)In(z)+(-1)n12(12z)nk=0(ψ(k+1)+ψ(n+k+1))(14z2)kk!(n+k)!,

where ψ(x)=Γ(x)/Γ(x)5.2(i)). In particular,

10.31.2 K0(z)=-(ln(12z)+γ)I0(z)+14z2(1!)2+(1+12)(14z2)2(2!)2+(1+12+13)(14z2)3(3!)2+.

For negative values of n use (10.27.3).

10.31.3 Iν(z)Iμ(z)=(12z)ν+μk=0(ν+μ+k+1)k(14z2)kk!Γ(ν+k+1)Γ(μ+k+1).