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HurwitzLerchPhi

HurwitzLerchPhi[z,s,a]

gives the Hurwitz–Lerch transcendent .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
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 .
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.
HurwitzLerchPhi can be used with Interval and CenteredInterval objects. »

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 (29)

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 (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 :

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 (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:

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