RamanujanTauZ

RamanujanTauZ[t]

gives the Ramanujan tau Z-function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • , where is the Ramanujan tau theta function, and is the Ramanujan tau L-function.
  • for real .
  • is an analytic function of except for branch cuts on the imaginary axis running from to .
  • For certain special arguments, RamanujanTauZ automatically evaluates to exact values.
  • RamanujanTauZ can be evaluated to arbitrary numerical precision.
  • RamanujanTauZ automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

Scope  (22)

Numerical Evaluation  (6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix RamanujanTauZ function using MatrixFunction:

Specific Values  (3)

Value at zero:

RamanujanTauZ evaluates to exact values for certain special arguments:

Find the first positive maximum of RamanujanTauZ[x]:

Visualization  (2)

Plot the RamanujanTauZ:

Plot the real part of the RamanujanTauZ function:

Plot the imaginary part of the RamanujanTauZ function:

Function Properties  (9)

RamanujanTauZ is defined for all real values:

Complex domain:

Approximate function range of RamanujanTauZ:

RamanujanTauZ threads over lists:

RamanujanTauZ is an analytic function of x:

RamanujanTauZ is neither non-increasing nor non-decreasing:

RamanujanTauZ is not injective:

RamanujanTauZ is neither non-negative nor non-positive:

RamanujanTauZ has no singularities or discontinuities:

RamanujanTauZ is neither convex nor concave:

Series Expansions  (2)

Find the Taylor expansion using Series:

Taylor expansion at a generic point:

Applications  (4)

Plot of the absolute value of RamanujanTauZ:

Find a numerical root:

On the critical line, RamanujanTauL splits:

Show interlacing of the roots of Sin[RamanujanTauTheta[t]] and RamanujanTauZ[t]]:

Properties & Relations  (3)

Relation to the Ramanujan tau L-function:

On the critical line, RamanujanTauZ is the modulus of RamanujanTauL up to a sign:

RamanujanTauZ can be expressed in terms of RamanujanTauL and RamanujanTauTheta:

Evaluate derivatives numerically:

Wolfram Research (2007), RamanujanTauZ, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauZ.html.

Text

Wolfram Research (2007), RamanujanTauZ, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauZ.html.

CMS

Wolfram Language. 2007. "RamanujanTauZ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTauZ.html.

APA

Wolfram Language. (2007). RamanujanTauZ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTauZ.html

BibTeX

@misc{reference.wolfram_2024_ramanujantauz, author="Wolfram Research", title="{RamanujanTauZ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauZ.html}", note=[Accessed: 21-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_ramanujantauz, organization={Wolfram Research}, title={RamanujanTauZ}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauZ.html}, note=[Accessed: 21-November-2024 ]}