WOLFRAM

LerchPhi[z,s,a]

gives the Lerch transcendent TemplateBox[{z, s, a}, LerchPhi].

Details and Options

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{z, s, a}, LerchPhi]=sum_(k=0)^(infty)z^k/(k+a)^s.
  • For , the definition used is TemplateBox[{z, s, a}, LerchPhi]=sum_(k=0)^(infty)z^k((k+a)^2)^(-s/2), 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.
  • LerchPhi can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (7)Summary of the most common use cases

Evaluate numerically:

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Simple exact values are generated automatically:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Series expansion at Infinity:

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Series expansion at a singular point:

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Scope  (29)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Evaluate to high precision:

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

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Complex number input:

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Evaluate efficiently at high precision:

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Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

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Compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix LerchPhi function using MatrixFunction:

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Specific Values  (7)

Simple exact values are generated automatically:

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

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

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

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Simple exact values are generated automatically:

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Values at zero:

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

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Visualization  (2)

Plot the LerchPhi function:

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Plot the real part of the LerchPhi function:

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Plot the imaginary part of the LerchPhi function:

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Function Properties  (11)

Real domain of LerchPhi:

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Complex domain:

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

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LerchPhi threads elementwise over lists and matrices:

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

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Nor is it meromorphic:

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

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

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

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

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

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

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TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to z:

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First derivative with respect to a:

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Higher derivatives with respect to z:

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

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Series Expansions  (1)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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Generalizations & Extensions  (2)Generalized and extended use cases

Series expansion at special points:

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LerchPhi can be applied to power series:

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Options  (4)Common values & functionality for each option

DoublyInfinite  (3)

By default, LerchPhi includes only terms with positive :

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

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

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IncludeSingularTerm  (1)

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

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Applications  (2)Sample problems that can be solved with this function

Find a zero of LerchPhi:

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Central moments of a geometric probability distribution:

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Explicit forms for small k:

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Properties & Relations  (2)Properties of the function, and connections to other functions

Obtain LerchPhi from sums:

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LerchPhi is a numeric function:

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Possible Issues  (4)Common pitfalls and unexpected behavior

A larger setting for $MaxExtraPrecision can be needed:

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LerchPhi uses numerical comparisons when singular terms are included:

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

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

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Wolfram Research (1988), LerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/LerchPhi.html (updated 2023).
Wolfram Research (1988), LerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/LerchPhi.html (updated 2023).

Text

Wolfram Research (1988), LerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/LerchPhi.html (updated 2023).

Wolfram Research (1988), LerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/LerchPhi.html (updated 2023).

CMS

Wolfram Language. 1988. "LerchPhi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LerchPhi.html.

Wolfram Language. 1988. "LerchPhi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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

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

BibTeX

@misc{reference.wolfram_2025_lerchphi, author="Wolfram Research", title="{LerchPhi}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LerchPhi.html}", note=[Accessed: 05-May-2025 ]}

@misc{reference.wolfram_2025_lerchphi, author="Wolfram Research", title="{LerchPhi}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LerchPhi.html}", note=[Accessed: 05-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_lerchphi, organization={Wolfram Research}, title={LerchPhi}, year={2023}, url={https://reference.wolfram.com/language/ref/LerchPhi.html}, note=[Accessed: 05-May-2025 ]}

@online{reference.wolfram_2025_lerchphi, organization={Wolfram Research}, title={LerchPhi}, year={2023}, url={https://reference.wolfram.com/language/ref/LerchPhi.html}, note=[Accessed: 05-May-2025 ]}