About the Project
NIST

psi function

AdvancedHelp

(0.002 seconds)

1—10 of 100 matching pages

1: 5.2 Definitions
§5.2(i) Gamma and Psi Functions
Euler’s Integral
5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
5.2.2 ψ ( z ) = Γ ( z ) / Γ ( z ) , z 0 , - 1 , - 2 , .
2: 5.1 Special Notation
j , m , n nonnegative integers.
The main functions treated in this chapter are the gamma function Γ ( z ) , the psi function (or digamma function) ψ ( z ) , the beta function B ( a , b ) , and the q -gamma function Γ q ( z ) . … Alternative notations for the psi function are: Ψ ( z - 1 ) (Gauss) Jahnke and Emde (1945); Ψ ( z ) Davis (1933); F ( z - 1 ) Pairman (1919).
3: 5.21 Methods of Computation
For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). Similarly for ln Γ ( z ) , ψ ( z ) , and the polygamma functions. …
4: 5.23 Approximations
§5.23(i) Rational Approximations
§5.23(ii) Expansions in Chebyshev Series
§5.23(iii) Approximations in the Complex Plane
See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of Γ ( z ) . …
5: 5.16 Sums
5.16.1 k = 1 ( - 1 ) k ψ ( k ) = - π 2 8 ,
5.16.2 k = 1 1 k ψ ( k + 1 ) = ζ ( 3 ) = - 1 2 ψ ′′ ( 1 ) .
For further sums involving the psi function see Hansen (1975, pp. 360–367). …
6: 5.22 Tables
§5.22(ii) Real Variables
§5.22(iii) Complex Variables
7: 5.3 Graphics
See accompanying text
Figure 5.3.3: ψ ( x ) . Magnify
See accompanying text
Figure 5.3.6: | ψ ( x + i y ) | . Magnify 3D Help
8: 5.5 Functional Relations
5.5.1 Γ ( z + 1 ) = z Γ ( z ) ,
5.5.2 ψ ( z + 1 ) = ψ ( z ) + 1 z .
5.5.3 Γ ( z ) Γ ( 1 - z ) = π / sin ( π z ) , z 0 , ± 1 , ,
5.5.7 k = 1 n - 1 Γ ( k n ) = ( 2 π ) ( n - 1 ) / 2 n - 1 / 2 .
5.5.8 ψ ( 2 z ) = 1 2 ( ψ ( z ) + ψ ( z + 1 2 ) ) + ln 2 ,
9: 5.7 Series Expansions
5.7.3 ln Γ ( 1 + z ) = - ln ( 1 + z ) + z ( 1 - γ ) + k = 2 ( - 1 ) k ( ζ ( k ) - 1 ) z k k , | z | < 2 .
5.7.4 ψ ( 1 + z ) = - γ + k = 2 ( - 1 ) k ζ ( k ) z k - 1 , | z | < 1 ,
5.7.5 ψ ( 1 + z ) = 1 2 z - π 2 cot ( π z ) + 1 z 2 - 1 + 1 - γ - k = 1 ( ζ ( 2 k + 1 ) - 1 ) z 2 k , | z | < 2 , z 0 , ± 1 .
§5.7(ii) Other Series
5.7.7 ψ ( z + 1 2 ) - ψ ( z 2 ) = 2 k = 0 ( - 1 ) k k + z .
10: 5.4 Special Values and Extrema
§5.4(ii) Psi Function
5.4.14 ψ ( n + 1 ) = k = 1 n 1 k - γ ,
5.4.19 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 ) ) .
§5.4(iii) Extrema
As n , …