§5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function (original) (raw)

Contents
  1. §5.2(i) Gamma and Psi Functions
  2. §5.2(ii) Euler’s Constant
  3. §5.2(iii) Pochhammer’s Symbol

§5.2(i) Gamma and Psi Functions

Euler’s Integral

5.2.1 Γ⁡(z)=∫0∞e−t⁢tz−1⁢dt,
ℜ⁡z>0.
ⓘ Defines: Γ⁡(z): gamma function Symbols: dx: differential of x,e: base of natural logarithm,∫: integral,ℜ⁡: real part andz: complex variable A&S Ref: 6.1.1 Referenced by: §10.43(iii),(25.11.27),(25.11.28),(25.5.6),(25.5.7),§5.9(i),§5.9(ii),§8.21(ii),(9.12.17) Permalink: http://dlmf.nist.gov/5.2.E1 Encodings: TeX, pMML, png See also: Annotations for §5.2(i),§5.2(i),§5.2 andCh.5

When ℜ⁡z≤0, Γ⁡(z) is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue (−1)n/n! at z=−n. 1/Γ⁡(z) is entire, with simple zeros at z=−n.

5.2.2 ψ⁡(z)=Γ′⁡(z)/Γ⁡(z),
z≠0,−1,−2,….
ⓘ Defines: ψ⁡(z): psi (or digamma) function Symbols: Γ⁡(z): gamma function andz: complex variable A&S Ref: 6.3.1 Permalink: http://dlmf.nist.gov/5.2.E2 Encodings: TeX, pMML, png See also: Annotations for §5.2(i),§5.2(i),§5.2 andCh.5

ψ⁡(z) is meromorphic with simple poles of residue −1 at z=−n.

§5.2(ii) Euler’s Constant

§5.2(iii) Pochhammer’s Symbol

5.2.4 (a)0 =1,
(a)n =a⁢(a+1)⁢(a+2)⁢⋯⁢(a+n−1),
ⓘ Symbols: (a)n: Pochhammer’s symbol (or shifted factorial),n: nonnegative integer anda: real or complex variable Referenced by: (25.11.10),(25.8.2) Permalink: http://dlmf.nist.gov/5.2.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.2(iii),§5.2 andCh.5
5.2.5 (a)n =Γ⁡(a+n)/Γ⁡(a),
a≠0,−1,−2,….
ⓘ Symbols: Γ⁡(z): gamma function,(a)n: Pochhammer’s symbol (or shifted factorial),n: nonnegative integer anda: real or complex variable A&S Ref: 6.1.22 Referenced by: (25.11.28),(25.5.7),(25.8.2) Permalink: http://dlmf.nist.gov/5.2.E5 Encodings: TeX, pMML, png See also: Annotations for §5.2(iii),§5.2 andCh.5
5.2.8 (a)2⁢n =22⁢n⁢(a2)n⁢(a+12)n,
(a)2⁢n+1 =22⁢n+1⁢(a2)n+1⁢(a+12)n.
ⓘ Symbols: (a)n: Pochhammer’s symbol (or shifted factorial),n: nonnegative integer anda: real or complex variable Referenced by: §5.2(iii),Erratum (V1.0.17) for Subsection 5.2(iii) Permalink: http://dlmf.nist.gov/5.2.E8 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.17): This equation was added.Suggested by Tom Koornwinder See also: Annotations for §5.2(iii),§5.2 andCh.5

Pochhammer symbols (rising factorials)(x)n=x⁢(x+1)⁢⋯⁢(x+n−1)and falling factorials(−1)n⁢(−x)n=x⁢(x−1)⁢⋯⁢(x−n+1) can be expressed in terms of each other via

5.2.9 (x)n =∑k=0nL⁡(n,k)⁢x⁢(x−1)⁢⋯⁢(x−k+1),
x⁢(x−1)⁢⋯⁢(x−n+1) =∑k=0n(−1)n−k⁢L⁡(n,k)⁢(x)k,
ⓘ Symbols: (a)n: Pochhammer’s symbol (or shifted factorial),n: nonnegative integer,k: nonnegative integer,x: real variable andL⁡(n,k): Lah number Referenced by: §5.2(iii),Erratum (V1.2.0) for Chapter 5 Addition Permalink: http://dlmf.nist.gov/5.2.E9 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was added.Suggested by Eric Shirley See also: Annotations for §5.2(iii),§5.2 andCh.5

in which L⁡(n,k)=(n−1k−1)⁢n!k! is the Lah number.