BUILT-IN MATHEMATICA SYMBOL

LerchPhi

gives the Lerch transcendent

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

.

For , the definition used is , where any term with is excluded.

LerchPhi[z, s, a, DoublyInfinite->True] gives the sum .

LerchPhi is a generalization of Zeta and PolyLog.

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

LerchPhi can be evaluated to arbitrary numerical precision.

LerchPhi automatically threads over lists.

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Scope (7)

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:

Generalizations & Extensions (2)

Series expansion at special points:

LerchPhi can be applied to power series:

Options (4)

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:

Applications (2)

Find a zero of LerchPhi:

Central moments of a geometric probability distribution:

Explicit forms for small

:

Properties & Relations (2)

Obtain LerchPhi from sums:

LerchPhi is a numeric function:

Possible Issues (5)

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: