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5 Gamma FunctionProperties

§5.4 Special Values and Extrema

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§5.4(i) Gamma Function

5.4.1 Γ(1) =1,
n! =Γ(n+1).
5.4.2 n!!={212nΓ(12n+1),n even,π-12212n+12Γ(12n+1),n odd.

(The second line of Formula (5.4.2) also applies when n=-1.)

5.4.3 |Γ(iy)|=(πysinh(πy))1/2,
5.4.4 Γ(12+iy)Γ(12-iy)=|Γ(12+iy)|2=πcosh(πy),
5.4.5 Γ(14+iy)Γ(34-iy)=π2cosh(πy)+isinh(πy).
5.4.6 Γ(12) =π1/2
=1.77245 38509 05516 02729,
5.4.7 Γ(13) =2.67893 85347 07747 63365,
5.4.8 Γ(23) =1.35411 79394 26400 41694,
5.4.9 Γ(14) =3.62560 99082 21908 31193,
5.4.10 Γ(34) =1.22541 67024 65177 64512.
5.4.11 Γ(1) =-γ.

§5.4(ii) Psi Function

5.4.12 ψ(1) =-γ,
ψ(1) =16π2,
5.4.13 ψ(12) =-γ-2ln2,
ψ(12) =12π2.

For higher derivatives of ψ(z) at z=1 and z=12, see §5.15.

5.4.14 ψ(n+1)=k=1n1k-γ,
5.4.15 ψ(n+12)=-γ-2ln2+2(1+13++12n-1),
n=1,2,.
5.4.16 ψ(iy)=12y+π2coth(πy),
5.4.17 ψ(12+iy)=π2tanh(πy),
5.4.18 ψ(1+iy)=-12y+π2coth(πy).

If p,q are integers with 0<p<q, then

5.4.19 ψ(pq)=-γ-lnq-π2cot(πpq)+12k=1q-1cos(2πkpq)ln(2-2cos(2πkq)).

§5.4(iii) Extrema

Table 5.4.1: Γ(xn)=ψ(xn)=0.
n xn Γ(xn)
0 1.46163 21449 0.88560 31944
1 -0.50408 30083 -3.54464 36112
2 -1.57349 84732 2.30240 72583
3 -2.61072 08875 -0.88813 63584
4 -3.63529 33665 0.24512 75398
5 -4.65323 77626 -0.05277 96396
6 -5.66716 24513 0.00932 45945
7 -6.67841 82649 -0.00139 73966
8 -7.68778 83250 0.00018 18784
9 -8.69576 41633 -0.00002 09253
10 -9.70267 25406 0.00000 21574

Compare Figure 5.3.1.

As n,

5.4.20 xn=-n+1πarctan(πlnn)+O(1n(lnn)2).

For error bounds for this estimate see Walker (2007, Theorem 5).