WOLFRAM

RamanujanTauTheta
Copy to clipboard.
RamanujanTauTheta

Copy to clipboard.

gives the Ramanujan tau theta function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • for real .
  • arises in the study of the Ramanujan L-function on the critical line. It is closely related to the number of zeros of for .
  • Apart from a sign, gives the phase of the Ramanujan L-function .
  • is an analytic function of except for branch cuts on the imaginary axis running from to .
  • For certain special arguments, RamanujanTauTheta automatically evaluates to exact values.
  • RamanujanTauTheta can be evaluated to arbitrary numerical precision.
  • RamanujanTauTheta automatically threads over lists.

Examples

open allclose all

Basic Examples  (6)Summary of the most common use cases

Evaluate numerically:

Out[1]=1

Plot over a subset of the reals:

Out[1]=1

Plot over a subset of the complexes:

Out[1]=1

Series expansion at the origin:

Out[1]=1

Series expansion at Infinity:

Out[1]=1

Series expansion at a singular point:

Out[1]=1

Scope  (27)Survey of the scope of standard use cases

Numerical Evaluation  (7)

Evaluate numerically:

Out[1]=1
Out[2]=2

Evaluate to high precision:

Out[1]=1
Out[2]=2

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

Out[1]=1

Complex number inputs:

Out[1]=1
Out[2]=2

Evaluate efficiently at high precision:

Out[1]=1
Out[2]=2

Compute average-case statistical intervals using Around:

Out[1]=1

Compute the elementwise values of an array:

Out[1]=1

Or compute the matrix RamanujanTauTheta function using MatrixFunction:

Out[2]=2

Specific Values  (2)

Value at zero:

Out[1]=1

Find positive minimum of RamanujanTauTheta[x]:

Out[1]=1
Out[2]=2

Visualization  (2)

Plot the RamanujanTauTheta:

Out[1]=1

Plot the real part of RamanujanTauTheta function:

Out[1]=1

Plot the imaginary part of RamanujanTauTheta function:

Out[2]=2

Function Properties  (10)

RamanujanTauTheta is defined for all real values:

Out[1]=1

Complex domain:

Out[2]=2

Function range of RamanujanTauTheta:

Out[1]=1

RamanujanTauTheta threads over lists:

Out[1]=1

RamanujanTauTheta is an analytic function of x:

Out[1]=1

RamanujanTauTheta is neither non-increasing nor non-decreasing:

Out[1]=1

RamanujanTauTheta is not injective:

Out[1]=1
Out[2]=2

RamanujanTauTheta is surjective:

Out[1]=1
Out[2]=2

RamanujanTauTheta is neither non-negative nor non-positive:

Out[1]=1

RamanujanTauTheta has no singularities or discontinuities:

Out[1]=1
Out[2]=2

RamanujanTauTheta is neither convex nor concave:

Out[1]=1

Differentiation  (2)

First derivative with respect to :

Out[1]=1

Higher derivatives with respect to :

Out[1]=1

Plot the higher derivatives with respect to :

Out[2]=2

Series Expansions  (4)

Find the Taylor expansion using Series:

Out[1]=1

Plots of the first three approximations around :

Out[3]=3

Find the series expansion at Infinity:

Out[1]=1

Find the series expansion for an arbitrary symbolic direction :

Out[1]=1

Taylor expansion at a generic point:

Out[1]=1

Applications  (4)Sample problems that can be solved with this function

Contour plot of the absolute value of RamanujanTauTheta:

Out[1]=1

The first 10 Gram points of RamanujanTauL:

Out[1]=1

Plot of RamanujanTauZ and Gram points:

Out[2]=2

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

Out[1]=1

The number of zeros on the critical strip from 0 to :

Out[2]=2

Properties & Relations  (3)Properties of the function, and connections to other functions

RamanujanTauTheta is related to LogGamma:

Out[1]=1
Out[2]=2

On the critical line, RamanujanTauTheta gives the phase of RamanujanTauL up to a sign:

Out[1]=1
Out[2]=2

RamanujanTauZ can be expressed in terms of RamanujanTauTheta and RamanujanTauL:

Out[1]=1
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Machine-number inputs can give high-precision results:

Out[1]=1
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Density plot of the argument:

Out[1]=1

Riemann surface of RamanujanTauTheta:

Out[1]=1
Wolfram Research (2007), RamanujanTauTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.
Copy to clipboard.
Wolfram Research (2007), RamanujanTauTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.

Text

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

Copy to clipboard.
Wolfram Research (2007), RamanujanTauTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.

CMS

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

Copy to clipboard.
Wolfram Language. 2007. "RamanujanTauTheta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2024_ramanujantautheta, author="Wolfram Research", title="{RamanujanTauTheta}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauTheta.html}", note=[Accessed: 08-January-2025 ]}

Copy to clipboard.
@misc{reference.wolfram_2024_ramanujantautheta, author="Wolfram Research", title="{RamanujanTauTheta}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauTheta.html}", note=[Accessed: 08-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_ramanujantautheta, organization={Wolfram Research}, title={RamanujanTauTheta}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauTheta.html}, note=[Accessed: 08-January-2025 ]}

Copy to clipboard.
@online{reference.wolfram_2024_ramanujantautheta, organization={Wolfram Research}, title={RamanujanTauTheta}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauTheta.html}, note=[Accessed: 08-January-2025 ]}