Dedekind eta function

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1: 23.16 Graphics

… ►See Figures 23.16.1–23.16.3 for the modular functions $\lambda$, $J$, and $\eta$. … ►

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Figure 23.16.1: Modular functions $\lambda \left(\mathrm{i}y\right)$, $J\left(\mathrm{i}y\right)$, $\eta \left(\mathrm{i}y\right)$ for $0\le y\le 3$. … Magnify

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Figure 23.16.3: Dedekind’s eta function $\eta \left(x+\mathrm{i}y\right)$ for $-0.0625\le x\le 0.0625$, $0.0001\le y\le 0.07$. Magnify 3D Help

2: 23.1 Special Notation

… ►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

3: 23.19 Interrelations

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23.19.1 $$\lambda \left(\tau \right)=16{\left(\frac{{\eta }^2\left(2\tau \right)\eta \left(\frac{1}{2}\tau \right)}{{\eta }^3\left(\tau \right)}\right)}^8,$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\lambda \left(\tau \right)$: elliptic modular function and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

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23.19.4 $$\mathrm{\Delta }={(2\pi )}^{12}{\eta }^{24}\left(\tau \right).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Delta }$: discriminant and $\tau$: complex variable

Referenced by:

§23.19

Permalink:

http://dlmf.nist.gov/23.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

4: 23.15 Definitions

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§23.15(ii) Functions $\lambda \left(\tau \right)$, $J\left(\tau \right)$, $\eta \left(\tau \right)$

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Dedekind’s Eta Function (or Dedekind Modular Function)

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23.15.9 $$\eta \left(\tau \right)={\left(\frac{1}{2}{\theta }_1^{\prime }(0,q)\right)}^{1/3}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}{\theta }_3\left(\frac{1}{2}\pi (1+\tau )\right|3\tau ).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), ${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nome and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.15(ii), §23.17(iii)

Permalink:

http://dlmf.nist.gov/23.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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5: 27.14 Unrestricted Partitions

… ►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by ►

27.14.12 $$\eta \left(\tau \right)={\mathrm{e}}^{\pi \mathrm{i}\tau /12}\prod _{n=1}^{\mathrm{\infty }}(1-{\mathrm{e}}^{2\pi \mathrm{i}n\tau }),$$ $\mathrm{\Im }\tau >0$.

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Defines:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $n$: positive integer

Referenced by:

§23.15(ii), §27.14(vi)

Permalink:

http://dlmf.nist.gov/27.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iv), §27.14 and Ch.27

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27.14.14 $$\eta \left(\frac{a\tau +b}{c\tau +d}\right)=\epsilon {(-\mathrm{i}(c\tau +d))}^{\frac{1}{2}}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§17.2(i)

Permalink:

http://dlmf.nist.gov/27.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iv), §27.14 and Ch.27

… ►For further properties of the function $\eta \left(\tau \right)$ see §§23.15–23.19. … ►

27.14.16 $$\mathrm{\Delta }\left(\tau \right)={(2\pi )}^{12}{(\eta \left(\tau \right))}^{24},$$ $\mathrm{\Im }\tau >0$,

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Defines:

$\mathrm{\Delta }\left(\tau \right)$: discriminant function

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{\Im }$: imaginary part

Permalink:

http://dlmf.nist.gov/27.14.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

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6: 20.3 Graphics

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Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. …Here $q^{\text{<span class="ltx\_text hilite">Dedekind</span>}}={\mathrm{e}}^{-\pi y_0}=0.19$ approximately, where $y=y_0$ corresponds to the maximum value of Dedekind’s eta function $\eta \left(\mathrm{i}y\right)$ as depicted in Figure 23.16.1. Magnify

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7: 23.17 Elementary Properties

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23.17.6 $$\eta \left(\tau \right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{{(6n+1)}^2/12}.$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

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23.17.8 $$\eta \left(\tau \right)=q^{1/12}\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n}),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.19

Permalink:

http://dlmf.nist.gov/23.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(iii), §23.17 and Ch.23

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8: 23.18 Modular Transformations

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Dedekind’s Eta Function

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23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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9: Bibliography M

… ►

Affine root systems and Dedekind’s $\eta$-function. Invent. Math. 15 (2), pp. 91–143.

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External Links:

Document, MathReview (D.-N. Verma), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

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10: 23.22 Methods of Computation

… ►The modular functions $\lambda \left(\tau \right)$, $J\left(\tau \right)$, and $\eta \left(\tau \right)$ are also obtainable in a similar manner from their definitions in §23.15(ii). …