HurwitzLerchPhi

HurwitzLerchPhi[z,s,a]

gives the HurwitzLerch transcendent TemplateBox[{z, s, a}, HurwitzLerchPhi].

Details

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  (28)

Numerical Evaluation  (5)

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:

HurwitzLerchPhi can be used with Interval and CenteredInterval objects:

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 function:

Plot the real part of the HurwitzLerchPhi function:

Plot the imaginary part of the HurwitzLerchPhi function:

Function Properties  (12)

Real domain of HurwitzLerchPhi:

Complex domain:

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

The defining sum for HurwitzLerchPhi:

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)

The moments and central moments of the geometric distribution can be expressed using HurwitzLerchPhi:

Explicit forms for the central moments for small k:

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 (updated 2023).

Text

Wolfram Research (2008), HurwitzLerchPhi, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html (updated 2023).

CMS

Wolfram Language. 2008. "HurwitzLerchPhi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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

BibTeX

@misc{reference.wolfram_2023_hurwitzlerchphi, author="Wolfram Research", title="{HurwitzLerchPhi}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html}", note=[Accessed: 13-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_hurwitzlerchphi, organization={Wolfram Research}, title={HurwitzLerchPhi}, year={2023}, url={https://reference.wolfram.com/language/ref/HurwitzLerchPhi.html}, note=[Accessed: 13-April-2024 ]}