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HurwitzLerchPhi

HurwitzLerchPhi[z,s,a]

gives the Hurwitz–Lerch transcendent .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
HurwitzLerchPhi is a generalization of Zeta[s], HurwitzZeta[s,a], PolyLog and related functions. »
The Hurwitz–Lerch transcendent is defined as an analytic continuation of $TemplateBox[{z, s, a}, HurwitzLerchPhi]=sum_(k=0)^(infty)z^k(k+a)^(-s)$ .
HurwitzLerchPhi is identical to LerchPhi for . »
HurwitzLerchPhi follows the branch cut conventions of the Hurwitz function as given by HurwitzZeta. By contrast, LerchPhi uses the branch cuts as defined by Zeta. »
Unlike LerchPhi, HurwitzLerchPhi has infinite or indeterminate values when the defining series has terms with zero denominator. »
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.
HurwitzLerchPhi can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (6)

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:

Scope (34)

Numerical Evaluation (6)

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:

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

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix HurwitzLerchPhi function using MatrixFunction:

Specific Values (6)

Simple exact values are generated automatically:

is a rational function in and a polynomial in if :

The following is manifestly a rational function in :

It is also a polynomial in :

HurwitzLerchPhi[z,s,1] is PolyLog[s,z]/z:

HurwitzLerchPhi[-1,s,a] gives expressions in HurwitzZeta:

HurwitzLerchPhi is indeterminate at the origin:

Approaching along the line gives :

Approaching the origin along the line also gives 1, but in a more interesting fashion:

Approaching along the line gives different results for and :

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

Visualization (3)

Plot the HurwitzLerchPhi function:

Plot the real part of the HurwitzLerchPhi function:

Plot the imaginary part of the HurwitzLerchPhi function:

Visualize the dependence on :

Visualize how LerchPhi and HurwitzLerchPhi agree for but not :

Function Properties (12)

Real domain of HurwitzLerchPhi:

Complex domain:

Function range of :

The defining sum for HurwitzLerchPhi:

HurwitzLerchPhi threads elementwise over lists:

is not an analytic function:

Nor is it meromorphic:

is neither non-decreasing nor non-increasing:

is injective:

is not surjective:

is neither non-negative nor non-positive:

has both singularity and discontinuity for x0 or for x≥1:

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 derivative with respect to a:

Series Expansions (4)

Find the Taylor expansion in for generic and using Series:

Plots of the first three approximations around :

Series expansion in at a generic point:

Series expansion about when has the singular value and :

Do the expansion about instead:

Series expansion in near , , :

Series expansion in about the same point:

HurwitzLerchPhi can be applied to power series:

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 (9)

Sum can generate HurwitzLerchPhi:

LerchPhi agrees with HurwitzLerchPhi for :

This is not true for :

HurwitzLerchPhi includes singular terms for which the denominator is zero:

The infinite value comes from the term in the defining series:

LerchPhi, by contrast, omits the singular term by default:

If , a singular term produces the value Indeterminate:

Zeta[s] equals HurwitzLerchPhi[1,s,1] for Re[s]>1:

HurwitzZeta[s,a] equals HurwitzLerchPhi[1,s,a] for Re[s]>1:

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

HurwitzLerchPhi matches HurwitzZeta, while LerchPhi matches Zeta:

PolyLog can be expressed in terms of HurwitzLerchPhi:

DirichletEta is a special case of HurwitzLerchPhi:

DirichletBeta is dilation of HurwitzLerchPhi:

Some hypergeometric functions can be expressed in terms of HurwitzLerchPhi:

Possible Issues (1)

The line is not considered to have a singular term:

This is consistent with Sum, which considers to be for all :

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