BUILT-IN MATHEMATICA SYMBOL

PolyGamma

PolyGamma[z]
gives the digamma function

.

PolyGamma

gives the n

derivative of the digamma function

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

PolyGamma[z] is the logarithmic derivative of the gamma function, given by .

PolyGamma is given for positive integer by .

For arbitrary complex n the polygamma function is defined by fractional calculus analytic continuation.

PolyGamma[z] and PolyGamma 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 (3)

Evaluate the digamma function:

Evaluate quadro-gamma:

Derivative of the gamma function:

The digamma function:

Evaluate the digamma function:

In[1]:=

Out[1]=

Evaluate quadro-gamma:

In[2]:=

Out[2]=

Derivative of the gamma function:

In[1]:=

Out[1]=

The digamma function:

In[1]:=

Out[1]=

Scope (9)

Evaluate for integer arguments of any size:

Evaluate numerically:

Evaluate for complex arguments and orders:

Evaluate to any precision:

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

PolyGamma threads element-wise over lists:

Use FunctionExpand to expand higher-order polygamma functions:

Series expansion:

TraditionalForm formatting:

Generalizations & Extensions (8)

Infinite arguments can give symbolic results:

PolyGamma can be applied to a power series:

Series expansion at poles:

Series expansion at infinities:

Special cases:

Evaluate at exact arguments:

Series expansion at a generic point:

Series expansion near a singularity:

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 one-argument form evaluates to the two-argument form:

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

Machine-number inputs can give high-precision results: