PolyGamma

PolyGamma[z]

gives the digamma function .

PolyGamma[n,z]

gives the n^(th) derivative of the digamma function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • PolyGamma[z] is the logarithmic derivative of the gamma function, given by .
  • PolyGamma[n,z] is given for positive integer by .
  • For arbitrary complex n, the polygamma function is defined by fractional calculus analytic continuation.
  • PolyGamma[z] and PolyGamma[n,z] are meromorphic functions of z with no branch cut discontinuities.
  • For certain special arguments, PolyGamma automatically evaluates to exact values.
  • PolyGamma can be evaluated to arbitrary numerical precision.
  • PolyGamma automatically threads over lists.

Examples

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Basic Examples  (7)

Evaluate the digamma function:

Evaluate quadrogamma:

Derivative of the gamma function:

Plot the digamma function over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope  (39)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate for integer arguments of any size:

Evaluate for complex arguments and orders:

Evaluate to any precision:

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

Evaluate PolyGamma efficiently at high precision:

PolyGamma threads elementwise over lists:

Specific Values  (6)

Some singular points of PolyGamma:

Values at infinity:

Find a zero of TemplateBox[{x}, PolyGamma]:

Use FunctionExpand to expand higher-order polygamma functions:

Special case:

Evaluate at exact arguments:

Visualization  (3)

Plot the digamma function:

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

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

Plot PolyGamma for half-integer values of parameter :

Function Properties  (2)

Real domain of PolyGamma:

Approximate function ranges of PolyGamma for half-integer parameters:

Differentiation  (3)

First derivative of PolyGamma:

Higher derivatives of Euler gamma function:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of PolyGamma:

Indefinite integral involving a power function:

Definite integral int_1^3TemplateBox[{x}, PolyGamma]dx:

Series Expansions  (7)

Taylor expansion for the digamma function around :

Plot the first three approximations for the Euler gamma function around :

General term in the series expansion of the digamma function:

Series expansion at infinities:

Series expansion at poles:

Series expansion at a generic point:

Series expansion near a singularity:

PolyGamma can be applied to a power series:

Function Identities and Simplifications  (5)

Use FullSimplify to simplify polygamma functions:

PolyGamma identity TemplateBox[{0, z}, PolyGamma2]=TemplateBox[{z}, PolyGamma]:

PolyGamma of a double argument:

Other argument simplifications:

Recurrence relation:

Function Representations  (4)

Digamma function definition:

Integral representation:

PolyGamma can be represented as a DifferenceRoot:

TraditionalForm formatting:

Applications  (3)

Plot of the absolute value of PolyGamma over the complex plane:

The electric field energy of a charge at a fraction of the distance between parallel conducting plates:

Expand near the left wall:

Final speed of a rocket with discrete propulsion events:

Final velocity in the limit of constant continuous propulsion:

Properties & Relations  (7)

Use FullSimplify to simplify polygamma functions:

Express rational arguments through elementary functions:

Numerically find a root of a transcendental equation:

Sums and integrals:

Generate PolyGamma from integrals, sums, and limits:

Generating function:

Obtain as special cases of hypergeometric functions:

Possible Issues  (3)

The oneargument form evaluates to the two-argument form:

Large orders can give results too large to be computed explicitly:

Machinenumber inputs can give highprecision results:

Introduced in 1988
 (1.0)
 |
Updated in 1999
 (4.0)
2000
 (4.1)
2002
 (4.2)
2007
 (6.0)