This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)


gives the Lerch transcendent .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • For , the definition used is , where any term with is excluded.
  • For certain special arguments, LerchPhi automatically evaluates to exact values.
  • LerchPhi can be evaluated to arbitrary numerical precision.
  • LerchPhi automatically threads over lists.
Click for copyable input
Click for copyable input
Click for copyable input
Evaluate for complex arguments and parameters:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
LerchPhi threads element-wise over lists and matrices:
Simple exact values are generated automatically:
Evaluate derivatives numerically:
TraditionalForm formatting:
Series expansion at special points:
LerchPhi can be applied to power series:
By default, LerchPhi includes only terms with positive :
In a symmetric case, setting True just doubles the result:
In a more general case, negative terms have a more complicated effect:
For negative integer a, True gives an infinite result:
Find a zero of LerchPhi:
Central moments of a geometric probability distribution:
Explicit forms for small :
Obtain LerchPhi from sums:
LerchPhi is a numeric function:
A larger setting for $MaxExtraPrecision can be needed:
LerchPhi uses numerical comparisons when singular terms are included:
Machine-number inputs can give high-precision results:
For , LerchPhi cannot always be evaluated in terms of Zeta for symbolic s:
HurwitzLerchPhi is different from LerchPhi in the choice of branch cuts:
New in 1