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10 Bessel FunctionsKelvin Functions

§10.65 Power Series

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§10.65(i) berνx and beiνx

10.65.1 berνx =(12x)νk=0cos(34νπ+12kπ)k!Γ(ν+k+1)(14x2)k,
beiνx =(12x)νk=0sin(34νπ+12kπ)k!Γ(ν+k+1)(14x2)k.
10.65.2 berx =1-(14x2)2(2!)2+(14x2)4(4!)2-,
beix =14x2-(14x2)3(3!)2+(14x2)5(5!)2-.

§10.65(ii) kerνx and keiνx

When ν is not an integer combine (10.65.1) with (10.61.6). Also, with ψ(x)=Γ(x)/Γ(x),

10.65.3 kernx =12(12x)-nk=0n-1(n-k-1)!k!cos(34nπ+12kπ)(14x2)k-ln(12x)bernx+14πbeinx+12(12x)nk=0ψ(k+1)+ψ(n+k+1)k!(n+k)!cos(34nπ+12kπ)(14x2)k,
10.65.4 keinx =-12(12x)-nk=0n-1(n-k-1)!k!sin(34nπ+12kπ)(14x2)k-ln(12x)beinx-14πbernx+12(12x)nk=0ψ(k+1)+ψ(n+k+1)k!(n+k)!sin(34nπ+12kπ)(14x2)k.
10.65.5 kerx =-ln(12x)berx+14πbeix+k=0(-1)kψ(2k+1)((2k)!)2(14x2)2k,
keix =-ln(12x)beix-14πberx+k=0(-1)kψ(2k+2)((2k+1)!)2(14x2)2k+1.

§10.65(iii) Cross-Products and Sums of Squares

10.65.6 berν2x+beiν2x=(12x)2νk=01Γ(ν+k+1)Γ(ν+2k+1)(14x2)2kk!,
10.65.7 berνxbeiνx-berνxbeiνx=(12x)2ν+1k=01Γ(ν+k+1)Γ(ν+2k+2)(14x2)2kk!,
10.65.8 berνxberνx+beiνxbeiνx=12(12x)2ν-1k=01Γ(ν+k+1)Γ(ν+2k)(14x2)2kk!,
10.65.9 (berνx)2+(beiνx)2=(12x)2ν-2k=02k2+2νk+14ν2Γ(ν+k+1)Γ(ν+2k+1)(14x2)2kk!.

§10.65(iv) Compendia

For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).