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5 Gamma FunctionProperties

§5.7 Series Expansions

  1. §5.7(i) Maclaurin and Taylor Series
  2. §5.7(ii) Other Series

§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 (k1)ck=γck1ζ(2)ck2+ζ(3)ck3+(1)kζ(k1)c1,

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(1γ)+k=2(1)k(ζ(k)1)zkk,
5.7.4 ψ(1+z) =γ+k=2(1)kζ(k)zk1,
5.7.5 ψ(1+z) =12zπ2cot(πz)+1z21+1γk=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)=γ1z+k=1zk(k+z)=γ+k=0(1k+11k+z),


5.7.7 ψ(z+12)ψ(z2)=2k=0(1)kk+z.


5.7.8 ψ(1+iy)=k=1yk2+y2.