Built-in Mathematica Symbol

LerchPhi

LerchPhi[z, s, a]
gives the Lerch transcendent

(z, s, a).

For certain special arguments, LerchPhi automatically evaluates to exact values.

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:

Series expansion at special points:

LerchPhi can be applied to power series:

By default, LerchPhi includes only terms with positive k:

In a symmetric case, setting DoublyInfinite->True just doubles the result:

In a more general case, negative

terms have a more complicated effect:

For negative integer a, IncludeSingularTerm->True gives an infinite result:

Find a zero of LerchPhi:

Central moments of a geometric probability distribution:

Explicit forms for small k:

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 z=a=1, LerchPhi cannot always be evaluated in terms of Zeta for symbolic s:

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