LogGamma

LogGamma[z]

gives the logarithm of the gamma function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • LogGamma[z] is analytic throughout the complex z plane, except for a single branch cut discontinuity along the negative real axis. Log[Gamma[z]] has a more complex branch cut structure.
  • For certain special arguments, LogGamma automatically evaluates to exact values.
  • LogGamma can be evaluated to arbitrary numerical precision.
  • LogGamma automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Evaluate at large arguments:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion about the origin:

Series expansion at Infinity:

Scope  (29)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Complex arguments:

Evaluate LogGamma efficiently at high precision:

LogGamma threads elementwise over lists:

Specific Values  (4)

Give exact results for integers and half-integers:

Some singular points of LogGamma:

Values at infinity:

Find a zero of LogGamma:

Visualization  (2)

Plot LogGamma:

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}}, LogGamma]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, LogGamma]:

Function Properties  (2)

LogGamma is defined for all positive real values:

Complex domain:

Differentiation  (3)

First derivative of LogGamma:

Higher derivatives of LogGamma:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of LogGamma:

Compute integrals involving LogGamma:

Definite integral for complex values of LogGamma:

Series Expansions  (5)

Series expansion at the origin:

Taylor expansion for LogGamma around :

Plot the first three approximations for LogGamma around :

Series expansion at infinity:

Give the result for an arbitrary symbolic direction :

Series expansion at poles of the LogGamma function:

LogGamma can be applied to a power series:

Function Identities and Simplifications  (3)

Use FullSimplify to simplify logarithmic gamma functions:

Use FunctionExpand to express through Gamma:

Recurrence relation:

Function Representations  (3)

Series representation of LogGamma:

Integral representation of LogGamma through PolyGamma:

TraditionalForm formatting:

Applications  (3)

Calculate ratio of Gamma functions at very large arguments:

Direct calculation fails because intermediate numbers are too large:

Find the first few digits of :

Plot of the imaginary part of LogGamma[z] and Log[Gamma[z]] over the complex plane:

Properties & Relations  (6)

Use FullSimplify to simplify logarithmic gamma functions:

Use FunctionExpand to express through Gamma:

Numerically find a root of a transcendental equation:

Check that TemplateBox[{x}, LogGamma] is convex for :

Find minimum of TemplateBox[{x}, LogGamma] on the positive real axis:

Visualize the result:

In TraditionalForm, is automatically interpreted as the gamma function:

Possible Issues  (2)

For many complex values :

Algorithmically generated results typically contain instead of :

Neat Examples  (2)

Plot LogGamma at the Gaussian integers:

Riemann surface of LogGamma:

Introduced in 1991
 (2.0)