gives the HurwitzLerch transcendent .


  • 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.


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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  (20)

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  (5)

Real domain of HurwitzLerchPhi:

Complex domain:

Function range of HurwitzLerchPhi:

HurwitzLerchPhi is defined through the identity:

HurwitzLerchPhi threads elementwise over lists:

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:

Introduced in 2008