LerchPhi

LerchPhi[z,s,a]

gives the Lerch transcendent Φ(z,s,a).

Details and Options

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • For , the definition used is , where any term with is excluded.
  • LerchPhi[z,s,a,DoublyInfinite->True] gives the sum .
  • LerchPhi is a generalization of Zeta and PolyLog.
  • For certain special arguments, LerchPhi automatically evaluates to exact values.
  • LerchPhi can be evaluated to arbitrary numerical precision.
  • LerchPhi automatically threads over lists.

Examples

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Basic Examples  (7)

Evaluate numerically:

Simple exact values are generated automatically:

Plot 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  (27)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (7)

Simple exact values are generated automatically:

LerchPhi[z,s,a] for symbolic a:

LerchPhi[z,s,a] for symbolic z:

LerchPhi[z,s,a] for symbolic s:

Simple exact values are generated automatically:

Values at zero:

Find a value of z for which LerchPhi[z,1,0]=1.05:

Visualization  (2)

Plot the LerchPhi:

Plot the real part of LerchPhi function:

Plot the imaginary part of LerchPhi function:

Function Properties  (11)

Real domain of LerchPhi:

Complex domain:

Approximate function range of TemplateBox[{x, {-, {1, /, 2}}, {-, 2}}, LerchPhi]:

LerchPhi threads elementwise over lists and matrices:

TemplateBox[{x, 1, 2}, LerchPhi] is not an analytic function:

Nor is it meromorphic:

TemplateBox[{x, 1, 2}, LerchPhi] is neither non-decreasing nor non-increasing:

TemplateBox[{x, 1, 2}, LerchPhi] is injective:

TemplateBox[{x, 1, 2}, LerchPhi] is not surjective:

TemplateBox[{x, 1, 2}, LerchPhi] is neither non-negative nor non-positive:

TemplateBox[{x, 1, 2}, LerchPhi] has both singularity and discontinuity for or for :

TemplateBox[{x, 1, 2}, LerchPhi] is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to z:

First derivative with respect to a:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when s=2 and a=1/3:

Series Expansions  (1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Generalizations & Extensions  (2)

Series expansion at special points:

LerchPhi can be applied to power series:

Options  (4)

DoublyInfinite  (3)

By default, LerchPhi includes only terms with positive :

In a symmetric case, setting DoublyInfinite->True just doubles the result:

In a more general case, negative terms have a more complicated effect:

IncludeSingularTerm  (1)

For negative integer a, IncludeSingularTerm->True gives an infinite result:

Applications  (2)

Find a zero of LerchPhi:

Central moments of a geometric probability distribution:

Explicit forms for small k:

Properties & Relations  (2)

Obtain LerchPhi from sums:

LerchPhi is a numeric function:

Possible Issues  (4)

A larger setting for $MaxExtraPrecision can be needed:

LerchPhi uses numerical comparisons when singular terms are included:

For z=a=1, LerchPhi cannot always be evaluated in terms of Zeta for symbolic s:

HurwitzLerchPhi is different from LerchPhi in the choice of branch cuts:

Wolfram Research (1988), LerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/LerchPhi.html.

Text

Wolfram Research (1988), LerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/LerchPhi.html.

BibTeX

@misc{reference.wolfram_2021_lerchphi, author="Wolfram Research", title="{LerchPhi}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/LerchPhi.html}", note=[Accessed: 29-July-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_lerchphi, organization={Wolfram Research}, title={LerchPhi}, year={1988}, url={https://reference.wolfram.com/language/ref/LerchPhi.html}, note=[Accessed: 29-July-2021 ]}

CMS

Wolfram Language. 1988. "LerchPhi." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LerchPhi.html.

APA

Wolfram Language. (1988). LerchPhi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LerchPhi.html