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5 Gamma FunctionProperties

§5.7 Series Expansions

Contents

§5.7(i) Maclaurin and Taylor Series

Throughout this subsection ζ(k) is as in Chapter 25.

5.7.1 1Γ(z)=k=1ckzk,

where c1=1, c2=γ, and

5.7.2 (k-1)ck=γck-1-ζ(2)ck-2+ζ(3)ck-3-+(-1)kζ(k-1)c1,
k3.

For 15D numerical values of ck see Abramowitz and Stegun (1964, p. 256), and for 31D values see Wrench (1968).

5.7.3 lnΓ(1+z) =-ln(1+z)+z(1-γ)+k=2(-1)k(ζ(k)-1)zkk,
|z|<2.
5.7.4 ψ(1+z) =-γ+k=2(-1)kζ(k)zk-1,
|z|<1,
5.7.5 ψ(1+z) =12z-π2cot(πz)+1z2-1+1-γ-k=1(ζ(2k+1)-1)z2k,
|z|<2, z0,±1.

For 20D numerical values of the coefficients of the Maclaurin series for Γ(z+3) see Luke (1969b, p. 299).

§5.7(ii) Other Series

When z0,-1,-2,,

5.7.6 ψ(z)=-γ-1z+k=1zk(k+z)=-γ+k=0(1k+1-1k+z),

and

5.7.7 ψ(z+12)-ψ(z2)=2k=0(-1)kk+z.

Also,

5.7.8 ψ(1+iy)=k=1yk2+y2.