Digital Library of Mathematical Functions
The Gamma Function -- Special Values and Extrema

Special Values and Extrema

Contents

Gamma Function

Γ ( 1 ) = 1
Γ ( n + 1 ) = n !
∣ Γ ( ⅈ y ) ∣ = ( π y sinh ( π y ) ) 1 2
Γ ( 1 2 + ⅈ y ) Γ ( 1 2 - ⅈ y ) = ∣ Γ ( 1 2 + ⅈ y ) ∣ 2 = π cosh ( π y )
Γ ( 1 4 + ⅈ y ) Γ ( 3 4 - ⅈ y ) = π 2 cosh ( π y ) + ⅈ sinh ( π y )
Γ ( 1 2 ) = π 1 2 = 1.77245 38509 05516 02729 …
Γ ( 1 3 ) = 2.67893 85347 07747 63365 …
Γ ( 2 3 ) = 1.35411 79394 26400 41694 …
Γ ( 1 4 ) = 3.62560 99082 21908 31193 …
Γ ( 3 4 ) = 1.22541 67024 65177 64512 …
Γ ′ ( 1 ) = - γ

Psi Function

ψ ( 1 ) = - γ
ψ ( 1 2 ) = - γ - 2 ln 2
ψ ( n + 1 ) = ∑ k = 1 n 1 k - γ
ψ ( n + 1 2 ) = - γ - 2 ln 2 + 2 ( 1 + 1 3 + … + 1 2 n - 1 ) n ≥ 1
ℑ ψ ( ⅈ y ) = 1 2 y + π 2 coth ( π y )
ℑ ψ ( 1 2 + ⅈ y ) = π 2 tanh ( π y )
ℑ ψ ( 1 + ⅈ y ) = - 1 2 y + π 2 coth ( π y )

0 < p < q are integers, then

ψ ( p q ) = - γ - ln q - π 2 cot ( π p q ) + 1 2 ∑ k = 1 q - 1 cos ( 2 π k p q ) ln ( 2 - 2 cos ( 2 π k q ) )

Extrema

Γ ′ ( x n ) = ψ ( x n ) = 0 .

n x n Γ ( x n )
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

As n ∞ ,

x n = - n + 1 π arctan ( π ln n ) + O ( 1 n ( ln n ) 2 )