RiemannXi

RiemannXi[s]

gives the Riemann xi function .

Details

  • Mathematical function, suitable for symbolic and numeric manipulations.
  • xi(s)=1/2 s (s-1) pi^(-s/2) TemplateBox[{s}, Zeta] TemplateBox[{{s, /, 2}}, Gamma].
  • For certain special arguments, RiemannXi automatically evaluates to exact values.
  • RiemannXi is an entire function with no branch cut discontinuities.
  • RiemannXi can be evaluated to arbitrary numerical precision.
  • RiemannXi automatically threads over lists.

Examples

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Basic Examples  (6)

Evaluate numerically:

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  (22)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (4)

Simple exact values are generated automatically:

Value at zero:

Evaluate symbolically:

Find the minimum of RiemannXi[x]:

Visualization  (2)

Plot the RiemannXi:

Plot the real part of the RiemannXi function:

Plot the imaginary part of the RiemannXi function:

Function Properties  (5)

Real domain of RiemannXi:

Complex domain:

RiemannXi has the mirror property TemplateBox[{{z, }}, RiemannXi]=TemplateBox[{z}, RiemannXi]:

RiemannXi is defined through the identity:

RiemannXi threads elementwise over lists:

TraditionalForm formatting, while avoiding the evaluation:

Differentiation  (3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Formula for the ^(th) derivative with respect to :

Series Expansions  (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Introduced in 2014
 (10.0)