RamanujanTauZ
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
open all close allBasic Examples (3)
Scope (22)
Numerical Evaluation (6)
RamanujanTauZ[5.]N[RamanujanTauZ[-14]]N[RamanujanTauZ[7 / 4], 50]N[RamanujanTauZ[2 / 3], 20]The precision of the output tracks the precision of the input:
RamanujanTauZ[21.21111111111111111111]N[RamanujanTauZ[3 - I]]N[RamanujanTauZ[4I]]Evaluate efficiently at high precision:
RamanujanTauZ[2`100]//TimingRamanujanTauZ[3`100];//TimingCompute average-case statistical intervals using Around:
RamanujanTauZ[ Around[2, 0.01]]Compute the elementwise values of an array:
RamanujanTauZ[{{I, -I}, {-I, I}}]Or compute the matrix RamanujanTauZ function using MatrixFunction:
MatrixFunction[RamanujanTauZ, {{I, -I}, {-I, I}}]//FullSimplifySpecific Values (3)
RamanujanTauZ[0]//N//QuietRamanujanTauZ evaluates to exact values for certain special arguments:
RamanujanTauZ[5 I]Find the first positive maximum of RamanujanTauZ[x]:
xmax = x /. FindRoot[D[RamanujanTauZ[x], x] == 0, {x, 5}]//QuietPlot[RamanujanTauZ[x], {x, 0, 20}, Epilog -> Style[Point[{xmax, RamanujanTauZ[xmax]}], PointSize[Large], Red]]//QuietVisualization (2)
Plot the RamanujanTauZ:
Plot[RamanujanTauZ[t], {t, 0, 20}]//QuietPlot the real part of the RamanujanTauZ function:
ContourPlot[Re[RamanujanTauZ[x + I y]], {x, -5, 5}, {y, -5, 5}, IconizedObject[«PlotOptions»]]//QuietPlot the imaginary part of the RamanujanTauZ function:
ContourPlot[Im[RamanujanTauZ[x + I y]], {x, -5, 5}, {y, -5, 5}, IconizedObject[«PlotOptions»]]//QuietFunction Properties (9)
RamanujanTauZ is defined for all real values:
FunctionDomain[RamanujanTauZ[x], x]FunctionDomain[RamanujanTauZ[z], z, Complexes]Approximate function range of RamanujanTauZ:
FunctionRange[RamanujanTauZ[x], x, y]//QuietRamanujanTauZ threads over lists:
RamanujanTauZ[{5.4, 9.0, 0.5}]RamanujanTauZ is an analytic function of x:
FunctionAnalytic[RamanujanTauZ[x], x]RamanujanTauZ is neither non-increasing nor non-decreasing:
FunctionMonotonicity[RamanujanTauZ[x], x]RamanujanTauZ is not injective:
FunctionInjective[RamanujanTauZ[x], x]Plot[{RamanujanTauZ[x], 1}, {x, 0, 20}]RamanujanTauZ is neither non-negative nor non-positive:
FunctionSign[RamanujanTauZ[x], x]RamanujanTauZ has no singularities or discontinuities:
FunctionSingularities[RamanujanTauZ[x], x]FunctionDiscontinuities[RamanujanTauZ[x], x]RamanujanTauZ is neither convex nor concave:
FunctionConvexity[RamanujanTauZ[x], x]Series Expansions (2)
Find the Taylor expansion using Series:
Series[RamanujanTauZ[x], {x, 0, 2}]Taylor expansion at a generic point:
Series[RamanujanTauZ[x], {x, x0, 2}]//Normal// FullSimplifyApplications (4)
Plot of the absolute value of RamanujanTauZ:
Plot3D[Abs[RamanujanTauZ[x + I y]], {x, -9, 9}, {y, -1, 1}]FindRoot[RamanujanTauZ[t], {t, 10}]Chop[RamanujanTauL[6 + I t /. %]]On the critical line, RamanujanTauL splits:
With[{s = 9.22`20}, RamanujanTauL[6 + I s] == RamanujanTauZ[s]Exp[-I RamanujanTauTheta[s]]]Show interlacing of the roots of Sin[RamanujanTauTheta[t]] and RamanujanTauZ[t]:
Plot[{Sin[RamanujanTauTheta[t]], RamanujanTauZ[t]}, {t, 0, 30}]Properties & Relations (3)
Relation to the Ramanujan tau L-function:
RamanujanTauL[6 + I t] - RamanujanTauZ[t]Exp[-I RamanujanTauTheta[t]]//FullSimplifyOn the critical line, RamanujanTauZ is the modulus of RamanujanTauL up to a sign:
Abs[RamanujanTauL[6 + 7.5I]]RamanujanTauZ[7.5]RamanujanTauZ can be expressed in terms of RamanujanTauL and RamanujanTauTheta:
With[{t = 7.5`20}, RamanujanTauZ[t] == Exp[I RamanujanTauTheta[t]] RamanujanTauL[6 + I t]]Evaluate derivatives numerically:
RamanujanTauZ'[9.22]RamanujanTauZ'''[9.22]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