HurwitzLerchPhi

HurwitzLerchPhi[z,s,a]

gives the HurwitzLerch transcendent .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The HurwitzLerch transcendent is defined as an analytic continuation of .
  • HurwitzLerchPhi is identical to LerchPhi for .
  • Unlike LerchPhi, HurwitzLerchPhi has singularities at for non-negative integers .
  • HurwitzLerchPhi has branch cut discontinuities in the complex plane running from to , and in the complex plane running from to .
  • For certain special arguments, HurwitzLerchPhi automatically evaluates to exact values.
  • HurwitzLerchPhi can be evaluated to arbitrary numerical precision.
  • HurwitzLerchPhi automatically threads over lists.

Examples

open allclose all

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  (5)

Simple exact values are generated automatically:

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

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

Values at zero:

Find a value of z for which HurwitzLerchPhi[z,1,1/2]=2.5:

Visualization  (2)

Plot the HurwitzLerchPhi:

Plot the real part of HurwitzLerchPhi function:

Plot the imaginary part of HurwitzLerchPhi function:

Function Properties  (12)

Real domain of HurwitzLerchPhi:

Complex domain:

Function range of TemplateBox[{x, 1, 2}, HurwitzLerchPhi]:

HurwitzLerchPhi is defined through the identity:

HurwitzLerchPhi threads elementwise over lists:

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

Nor is it meromorphic:

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

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

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

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

TemplateBox[{x, 1, 2}, HurwitzLerchPhi] has both singularity and discontinuity for x0 or for x1:

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

TraditionalForm formatting:

Differentiation  (3)

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 a=5 and s=-1/2:

Formula for the ^(th) derivative with respect to a:

Series Expansions (1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Applications  (1)

Moment and central moment of geometric distribution can be expressed using HurwitzLerchPhi:

Properties & Relations  (2)

Some hypergeometric functions can be expressed in terms of HurwitzLerchPhi:

Sum can generate HurwitzLerchPhi:

Possible Issues  (2)

HurwitzLerchPhi differs from LerchPhi by a different choice of branch cut:

HurwitzLerchPhi includes singular terms, unlike LerchPhi:

Wolfram Research (2008), HurwitzLerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html.

Text

Wolfram Research (2008), HurwitzLerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html.

BibTeX

@misc{reference.wolfram_2021_hurwitzlerchphi, author="Wolfram Research", title="{HurwitzLerchPhi}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html}", note=[Accessed: 22-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hurwitzlerchphi, organization={Wolfram Research}, title={HurwitzLerchPhi}, year={2008}, url={https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html}, note=[Accessed: 22-September-2021 ]}

CMS

Wolfram Language. 2008. "HurwitzLerchPhi." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html.

APA

Wolfram Language. (2008). HurwitzLerchPhi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html