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5 Gamma FunctionProperties

§5.4 Special Values and Extrema

Contents
  1. §5.4(i) Gamma Function
  2. §5.4(ii) Psi Function
  3. §5.4(iii) Extrema

§5.4(i) Gamma Function

5.4.1 Γ(1) =1,
n! =Γ(n+1).
5.4.2 n!!={212nΓ(12n+1),n even,π12212n+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(12iy)=|Γ(12+iy)|2=πcosh(πy),
5.4.5 Γ(14+iy)Γ(34iy)=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) =γ2ln2,
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=1n1kγ,
5.4.15 ψ(n+12)=γ2ln2+2(1+13++12n1),
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)=γlnqπ2cot(πpq)+12k=1q1cos(2πkpq)ln(22cos(2πkq)).

§5.4(iii) Extrema

Table 5.4.1: Γ(xn)=ψ(xn)=0.
n xn Γ(xn)
0 1.46163 21449 68362 34126 0.88560 31944 10888 70028
1 0.50408 30082 64455 40926 3.54464 36111 55005 08912
2 1.57349 84731 62390 45878 2.30240 72583 39680 13582
3 2.61072 08684 44144 65000 0.88813 63584 01241 92010
4 3.63529 33664 36901 09784 0.24512 75398 34366 25044
5 4.65323 77617 43142 44171 0.05277 96395 87319 40076
6 5.66716 24415 56885 53585 0.00932 45944 82614 85052
7 6.67841 82130 73426 74283 0.00139 73966 08949 76730
8 7.68778 83250 31626 03744 0.00018 18784 44909 40419
9 8.69576 41638 16401 26649 0.00002 09252 90446 52667
10 9.70267 25400 01863 73608 0.00000 21574 16104 52285

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).