LerchPhi
✖
LerchPhi
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.
- LerchPhi can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic Examples (7)Summary of the most common use cases

https://wolfram.com/xid/01zbv5hv6r3-g0m

Simple exact values are generated automatically:

https://wolfram.com/xid/01zbv5hv6r3-yjd

Plot over a subset of the reals:

https://wolfram.com/xid/01zbv5hv6r3-yps

Plot over a subset of the complexes:

https://wolfram.com/xid/01zbv5hv6r3-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/01zbv5hv6r3-fdkkja

Series expansion at Infinity:

https://wolfram.com/xid/01zbv5hv6r3-20imb

Series expansion at a singular point:

https://wolfram.com/xid/01zbv5hv6r3-d2klx1

Scope (29)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/01zbv5hv6r3-l274ju


https://wolfram.com/xid/01zbv5hv6r3-cksbl4


https://wolfram.com/xid/01zbv5hv6r3-b0wt9

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

https://wolfram.com/xid/01zbv5hv6r3-y7k4a


https://wolfram.com/xid/01zbv5hv6r3-k6bv9

Evaluate efficiently at high precision:

https://wolfram.com/xid/01zbv5hv6r3-di5gcr


https://wolfram.com/xid/01zbv5hv6r3-bq2c6r

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

https://wolfram.com/xid/01zbv5hv6r3-bo9lc8


https://wolfram.com/xid/01zbv5hv6r3-mfxf9l

Compute average-case statistical intervals using Around:

https://wolfram.com/xid/01zbv5hv6r3-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/01zbv5hv6r3-thgd2

Or compute the matrix LerchPhi function using MatrixFunction:

https://wolfram.com/xid/01zbv5hv6r3-o5jpo

Specific Values (7)
Simple exact values are generated automatically:

https://wolfram.com/xid/01zbv5hv6r3-nww7l

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

https://wolfram.com/xid/01zbv5hv6r3-fc9m8o

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

https://wolfram.com/xid/01zbv5hv6r3-h7p5ce

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

https://wolfram.com/xid/01zbv5hv6r3-o7j

Simple exact values are generated automatically:

https://wolfram.com/xid/01zbv5hv6r3-jzp


https://wolfram.com/xid/01zbv5hv6r3-g55


https://wolfram.com/xid/01zbv5hv6r3-bmqd0y

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

https://wolfram.com/xid/01zbv5hv6r3-f2hrld


https://wolfram.com/xid/01zbv5hv6r3-fvjkvq

Visualization (2)
Plot the LerchPhi function:

https://wolfram.com/xid/01zbv5hv6r3-b1j98m

Plot the real part of the LerchPhi function:

https://wolfram.com/xid/01zbv5hv6r3-d25e0g

Plot the imaginary part of the LerchPhi function:

https://wolfram.com/xid/01zbv5hv6r3-qbv4xx

Function Properties (11)
Real domain of LerchPhi:

https://wolfram.com/xid/01zbv5hv6r3-pri393


https://wolfram.com/xid/01zbv5hv6r3-fv671y

Approximate function range of :

https://wolfram.com/xid/01zbv5hv6r3-evf2yr

LerchPhi threads elementwise over lists and matrices:

https://wolfram.com/xid/01zbv5hv6r3-m5e


https://wolfram.com/xid/01zbv5hv6r3-h5x4l2


https://wolfram.com/xid/01zbv5hv6r3-eczogh

is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/01zbv5hv6r3-g6kynf


https://wolfram.com/xid/01zbv5hv6r3-gi38d7


https://wolfram.com/xid/01zbv5hv6r3-ctca0g


https://wolfram.com/xid/01zbv5hv6r3-hkqec4


https://wolfram.com/xid/01zbv5hv6r3-hdm869

is neither non-negative nor non-positive:

https://wolfram.com/xid/01zbv5hv6r3-84dui

has both singularity and discontinuity for
or for
:

https://wolfram.com/xid/01zbv5hv6r3-mdtl3h


https://wolfram.com/xid/01zbv5hv6r3-mn5jws

is neither convex nor concave:

https://wolfram.com/xid/01zbv5hv6r3-kdss3

TraditionalForm formatting:

https://wolfram.com/xid/01zbv5hv6r3-g5hh1q

Differentiation (2)
First derivative with respect to z:

https://wolfram.com/xid/01zbv5hv6r3-krpoah

First derivative with respect to a:

https://wolfram.com/xid/01zbv5hv6r3-g5o178

Higher derivatives with respect to z:

https://wolfram.com/xid/01zbv5hv6r3-z33jv

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

https://wolfram.com/xid/01zbv5hv6r3-fxwmfc

Series Expansions (1)
Find the Taylor expansion using Series:

https://wolfram.com/xid/01zbv5hv6r3-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/01zbv5hv6r3-binhar

Generalizations & Extensions (2)Generalized and extended use cases
Series expansion at special points:

https://wolfram.com/xid/01zbv5hv6r3-k3t


https://wolfram.com/xid/01zbv5hv6r3-vh2

LerchPhi can be applied to power series:

https://wolfram.com/xid/01zbv5hv6r3-r2c

Options (4)Common values & functionality for each option
DoublyInfinite (3)
By default, LerchPhi includes only terms with positive :

https://wolfram.com/xid/01zbv5hv6r3-yxk

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

https://wolfram.com/xid/01zbv5hv6r3-vm4


https://wolfram.com/xid/01zbv5hv6r3-uag

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

https://wolfram.com/xid/01zbv5hv6r3-ci5


https://wolfram.com/xid/01zbv5hv6r3-xrt

IncludeSingularTerm (1)
For negative integer a, IncludeSingularTerm->True gives an infinite result:

https://wolfram.com/xid/01zbv5hv6r3-fm0


https://wolfram.com/xid/01zbv5hv6r3-jc5

Applications (2)Sample problems that can be solved with this function
Find a zero of LerchPhi:

https://wolfram.com/xid/01zbv5hv6r3-u3q

Central moments of a geometric probability distribution:

https://wolfram.com/xid/01zbv5hv6r3-kic


https://wolfram.com/xid/01zbv5hv6r3-q7

Properties & Relations (2)Properties of the function, and connections to other functions
Possible Issues (4)Common pitfalls and unexpected behavior
A larger setting for $MaxExtraPrecision can be needed:

https://wolfram.com/xid/01zbv5hv6r3-jfz




https://wolfram.com/xid/01zbv5hv6r3-wfi

LerchPhi uses numerical comparisons when singular terms are included:

https://wolfram.com/xid/01zbv5hv6r3-xrj


https://wolfram.com/xid/01zbv5hv6r3-esa

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

https://wolfram.com/xid/01zbv5hv6r3-m3p


https://wolfram.com/xid/01zbv5hv6r3-k6x


https://wolfram.com/xid/01zbv5hv6r3-x88


https://wolfram.com/xid/01zbv5hv6r3-xx8

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

https://wolfram.com/xid/01zbv5hv6r3-ljn638


https://wolfram.com/xid/01zbv5hv6r3-g0q7r

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
]}
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
]}