Gamma

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Γ(z) is an analytical function of z which is defined over the whole complex z-plane with the exception of countably many points z=-k/;k∈ℕ. 1/Γ(z) is an entire function.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

Formulas for residues of ratios of products of gamma functions such as (Subscript[res, s])(((((∏_(k=1)^A)Γ(s+(Subscript[a, k])))(∏_(k=1)^B)Γ((Subscript[b, k])-s))/(((∏_(k=1)^C)Γ(s+(Subscript[c, k])))(∏_(k=1)^D)Γ((Subscript[d, k])-s)))(z^(-s)))(-(Subscript[a, j])-l) in regular and logarithmic cases are represented in the subsection Residues of ratios below.

Branch points

The function Γ(z) does not have branch points.

Branch cuts

The function Γ(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

Expansions at z==(Subscript[z, 0])/; (Subscript[z, 0])≠-n

Expansions at z==-n

Expansions of Gamma[z+ϵ] at ϵ==0/;z≠-n

Expansions of Gamma[-n+ϵ] at ϵ==0

Expansions of 1/Gamma[z]

Asymptotic series expansions

Integral representations

On the real axis

Of the direct function

A. Radović

A. Radović

Contour integral representations

Product representations

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

The gamma function does not satisfy any algebraic differential equation (O.Hölder, 1887). But it is the solution of the following nonalgebraic equation:

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Γ(x) is the unique nonzero solution of the functional equation f(x+1)==x f(x)
which is logarithmically convex for all real x>0, that is, for which log(f(x)) is a convex function for x>0.

Γ(x) is the unique non-zero continuously differentiable solution of the system of functional equations
	  f(2x)==((2^(2 x-1))/(Sqrt[π]))f(x)f(x+(1/2)) and f(x+1)==x f(x).

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Definite integration

Involving the direct function

Integral transforms

Inverse Mellin transforms

Summation

Infinite summation

Products

Finite products

Operations

Limit operation

Residues of ratios of gamma functions

Case of simple poles

Case of double poles

Case of triple poles

Case of quartic poles

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Inequalities

Zeros

Theorems

Riemann-Liouville fractional integration

(Overscript[(Subscript[f,  α]), ⋀])(y)==(∫_c^y)f(x)((((y-x))^(α-1))/(Γ(α)))dx⇔ f(x)==(1/(Γ(n-α)))((∂^n)/(∂(x^n)))(∫_c^x)(Overscript[(Subscript[f,  α]), ⋀])(y)(((x-y))^(n-α-1))dy/;n-1≤α<n∧n∈(SuperPlus[ℕ])

The value of the Barnes integral

(∫_(γ-i ∞)^(γ+i ∞))Γ(a+s) Γ(b+s) Γ(c-s) Γ(d-s)ds==((Γ(a+c) Γ(a+d) Γ(b+c) Γ(b+d))/(Γ(a+b+c+d)))/;-Re(a),-Re(b)<γ<Re(c),Re(d).

The scattering amplitude in the outgoing wave function

The scattering amplitude f(k;n',n) in the outgoing wave function

ψ(r)(Underscript[∼, x->∞])(e^(i k r+i/k log(k r -k r n'.n)))+(1/r)(e^(i k r-i/k log(2k r)))f(k;n',n)

for the Coulomb potential V(r)=(α/r) after renormalization is given by

f(k;n',n)==(1/(2i k))(∑_(l=0)^∞)(2l+1)((Γ(l+1+i /k))/(Γ(l+1-i /k)))(Subscript[P, l])(n'.n)==((i )/(2(k^2)))(((4|n-n'((|)^(-2))))^(1+i /k))((Γ(1+i /k))/(Γ(1-i /k)))

where η == η(k).

Selberg's integral

(∫_0^1)(∫_0^1)…(∫_0^1)|(∏_(Underscript[j,l=1, j<l])^n)((Subscript[x, j])-(Subscript[x, l]))((|)^(2γ))(∏_(k=1)^n)((((Subscript[x, k]^(α-1))(1-(Subscript[x, k]))))^(β-1))d(Subscript[x, 1])d(Subscript[x, 2])…d(Subscript[x, n])== (∏_(k=0)^n)((Γ(1+γ+k γ)Γ(α+k γ)Γ(β+k γ))/(Γ(1+γ)Γ(α+β+(n+k-1) γ)))/;                      
 Re(α),Re(β)>0,Re(γ)>-min((1/n),((Re(α))/(n-1)),((Re(b))/(n-1))).

Dirichlet integral

(Underscript[(∫_0^1)(∫_0^1)…(∫_0^1), (Subscript[x, 1])+(Subscript[x, 2])+…+(Subscript[x, n])≤1])f((∑_(k=1)^n)(Subscript[x, k]) )(∏_(k=1)^n)(Subscript[x, k]^((Subscript[α, k])-1))d(Subscript[x, 1])d(Subscript[x, 2])…d(Subscript[x, n])==(∏_(k=1)^n)Γ((Subscript[α, k]))/Γ((∑_(k=1)^n)(Subscript[α, k]))(∫_0^1)f(x)(x^((∑_(k=1)^n)(Subscript[α, k])-1))dx/;Re((Subscript[α, k]))>0,k==1,2,…,n

Another Dirichlet-type integral

(Underscript[(Underscript[∫…∫, (Subscript[x, 1]),(Subscript[x, 2]),…,(Subscript[x, n>0])]), (Subscript[x, 1])+(Subscript[x, 2])+…+(Subscript[x, n])≤1])(Subscript[x, 1]^((Subscript[α, 1])-1))(Subscript[x, 2]^((Subscript[α, 2])-1))(Subscript[…x, n]^((Subscript[α, n])-1))d(Subscript[x, 1])d(Subscript[x, 2])…d(Subscript[x, n])==((Γ((Subscript[α, 1]))Γ((Subscript[α, 2]))…Γ((Subscript[α, n])))/(Γ((Subscript[α, 1])+…+(Subscript[α, n])+1))).

The micro-canonical single particle distribution function for a multi-dimensional ideal gas

The micro-canonical single particle distribution function p(z,v) for an f-dimensional ideal gas in a gravitational field with acceleration g is given by

p(z,v)==(((2^((2-f)/2))Γ( (f+2) N/2))/(Γ(f/2) Γ( (f+2) (N-1)/2))) (g (m^((2-f)/2))  (v^(f-1))(ε^((2-f)/2))) (((-((m (v^2))/(2 ε))-((m g z)/ε)+1))^( ((f+2) N-(f+4))/2)),

where m is the particle mass and  ε its energy.

History

–  L. Euler (1729) derived integral representation for n!, leading to gamma function
–  J. Stirling (1730) found main asymptotical term for n!
– A. M. Legendre (1808, 1814) suggested the notation Γ and discovered known duplication formula
– C. F. Gauss (1812) derived the multiplication formula
– F.W. Newman (1848) found product representation for the reciprocal of the gamma function that are valid for whole complex plane
– B. Riemann (1856) proved important relation between gamma and zeta functions
- K. Weierstrass (1856) widely used gamma function in his inverrstigations
– H. Hankel (1864, 1880) derived a contour integral representation for complex arguments
– O. Hölder (1887) proved that gamma function does not satisfy any algebraic differential equation
– H. Bohr and J. Mollerup (1922) proved that the gamma function is the only function that satisfies the recurrence relationship and is logarithmically convex

Encountered often in mathematics and the natural sciences.

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