LogGamma

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

logΓ(z) is an analytical function of z which is defined over the whole complex z-plane. It has one infinitely long branch cut.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function logΓ(z) does not have poles and essential singularities.

Branch points

Branch cuts

The function  logΓ(z) is a single-valued function on the z-plane cut along the interval (-∞, 0), where  logΓ(z) is continuous from above. This interval includes an infinite set of branch cut lines of combined logarithmic type along (-∞, -n)/;n∈ℕ.

Series representations

Generalized power series

Expansions on branch cuts

Expansions at z==0

Expansions at z==(Subscript[z, 0])/; ¬((Subscript[z, 0])∈ℝ∧(Subscript[z, 0])≤0)

Expansions at z==-n

Exponential Fourier series

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

Involving the direct function

Operations

Limit operation

Representations through more general functions

Through other functions

Representations through equivalent functions

With related functions

History

–  J. Stirling (1730) used series for log(n!) to derive his famous asymptotic formula
– C. Siegel 
– K. F. Gauss
– C. J. Malmstén
– O. Schlömilch
– J. P. M. Binet (1843)
– E. E. Kummer (1847)
– G. Plana (1847)
– M.A. Stern (1847) proved convergence of the Stirling series for derivative of log(Γ(z))
– Ch. Hermite (1900) proved convergence of Stirling's series for log(Γ(z+1)) to this function if z is a complex number
– J. Keiper (1990) (introduced logΓ(z) in Mathematica)

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