![TemplateBox[{tau}, DedekindEta]](Files/DedekindEta.en/19.png "TemplateBox[{tau}, DedekindEta]"){kind=small}
DedekindEta
DedekindEta[τ]
gives the Dedekind eta modular elliptic function ![TemplateBox[{tau}, DedekindEta]](Files/DedekindEta.en/1.png "TemplateBox[{tau}, DedekindEta]"){kind=small}
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- DedekindEta is defined only in the upper half of the complex τ plane. It is not defined for real τ.
- The argument τ is the ratio of Weierstrass half‐periods
. - DedekindEta satisfies
where 
is the discriminant, given in terms of Weierstrass invariants by 
. - For certain special arguments, DedekindEta automatically evaluates to exact values.
- DedekindEta can be evaluated to arbitrary numerical precision.
- DedekindEta automatically threads over lists.
- DedekindEta can be used with CenteredInterval objects. »
Examples
open all close allBasic Examples (2)
Scope (14)
Numerical Evaluation (4)
DedekindEta[14.3 + I]DedekindEta[1.5 + I]N[DedekindEta[12 / 5 + I], 50]The precision of the output tracks the precision of the input:
DedekindEta[2.33333333333333333 + I]Evaluate efficiently at high precision:
DedekindEta[I + 1 / 7`100]//TimingDedekindEta[I + 1 / 16`10000];//TimingDedekindEta can be used with CenteredInterval objects:
DedekindEta[CenteredInterval[-4 + I, .2]]Specific Values (2)
DedekindEta[I]DedekindEta threads elementwise over lists:
DedekindEta[{0.2 + I, 0.3 + 2I, 0.4 + 3I}]Visualization (2)
Plot the DedekindEta function for various parameters:
Plot[Re[DedekindEta[x + I]], {x, -12, 12}]Plot the real part of the DedekindEta function in three dimensions:
Plot3D[Re[DedekindEta[x + I y]], {x, -2, 2}, {y, -2, 2}, ColorFunction -> "BlueGreenYellow", PlotRange -> All]//QuietPlot the imaginary part of the DedekindEta function in three dimensions:
Plot3D[Im[DedekindEta[x + I y]], {x, -4, 4}, {y, -4, 4}, ColorFunction -> "BlueGreenYellow", PlotRange -> All]//QuietFunction Properties (6)
Complex domain of DedekindEta:
FunctionDomain[DedekindEta[x], x, Complexes]DedekindEta is a periodic function:
FunctionPeriod[DedekindEta[x + I], x]DedekindEta is an analytic function on its domain:
FunctionAnalytic[{DedekindEta[x], Im[x] > 0}, x, Complexes]It is not an entire function, however:
FunctionAnalytic[DedekindEta[x], x, Complexes]It has both singularities and discontinuities:
FunctionSingularities[DedekindEta[x], x, Complexes]FunctionDiscontinuities[DedekindEta[x], x, Complexes]DedekindEta is not injective over the complexes:
FunctionInjective[DedekindEta[x + 0.2I], x, Complexes]Plot[{Re[DedekindEta[x + 0.2I]], 1}, {x, -6, 6}]DedekindEta is not surjective:
FunctionSurjective[Re[DedekindEta[x + 0.2I]], x]Plot[{Re[DedekindEta[x + 0.2I]], -2}, {x, -6, 6}]TraditionalForm formatting:
DedekindEta[τ]//TraditionalFormApplications (3)
The modular discriminant at I is given by DedekindEta:
(2 Pi) ^ 12 DedekindEta[I] ^ 24//NCompare with the general definition:
m[q_] := (2 Pi) ^ 12 NSum[
RamanujanTau[n] E ^ (2 Pi I n q), {n, 1, Infinity}]
m[I]Plot the DedekindEta function in the upper half of the complex plane:
ContourPlot[Re[DedekindEta[x + I y]], {x, -1, 1}, {y, 0.01, 2}, AspectRatio -> Automatic]m[τ_] := (2Pi) ^ 12 NSum[RamanujanTau[n] (E^2π I τ)^n, {n, 1, Infinity}]Relation with DedekindEta:
DedekindEta[2I]^24 == m[2I] / (2 Pi) ^ 12Properties & Relations (2)
Machine-precision input is insufficient to give a correct answer:
DedekindEta[10. ^ 22 Pi + I ]With exact input, the answer is correct:
N[DedekindEta[10 ^ 22 + I ], 20]Because DedekindEta is a numerical function with numeric arguments, it might be considered a numeric quantity but because of its boundary of analyticity, it might not be evaluatable to a number:
DedekindEta[Pi]NumericQ[%]N[%%]Related Links
MathWorld
The Wolfram Functions Site
NKS|OnlineText
Wolfram Research (1996), DedekindEta, Wolfram Language function, https://reference.wolfram.com/language/ref/DedekindEta.html (updated 2021).
CMS
Wolfram Language. 1996. "DedekindEta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/DedekindEta.html.
APA
Wolfram Language. (1996). DedekindEta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DedekindEta.html