20 Theta Functions Properties 20.1 Special Notation 20.3 Graphics

§20.2 Definitions and Periodic Properties

ⓘ

Permalink:: http://dlmf.nist.gov/20.2
See also:: Annotations for Ch.20

§20.2(i) Fourier Series

ⓘ

Defines:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function
Keywords:: Fourier series, Jacobi’s theta functions, classical theta functions, multidimensional, theta functions
Notes:: See Whittaker and Watson (1927, p. 464) or Lawden (1989, p. 5).
Referenced by:: §20.14, §21.2(iii), §22.16(ii), §22.2, §23.15(ii), §23.17(ii), §25.5(ii), §27.13(iv)
Permalink:: http://dlmf.nist.gov/20.2.i
See also:: Annotations for §20.2 and Ch.20

20.2.1	$\displaystyle\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{1}\left(z,q\right)=2\sum_{n=0}^{\infty}(-1)^{n}q^{(n+% \frac{1}{2})^{2}}\sin\left((2n+1)z\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\sin\NVar{z}$ : sine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.27.1 Referenced by: §20.14, §20.2(i) Permalink: http://dlmf.nist.gov/20.2.E1 Encodings: TeX, pMML, png See also: Annotations for §20.2(i), §20.2 and Ch.20
20.2.2	$\displaystyle\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{2}\left(z,q\right)=2\sum_{n=0}^{\infty}q^{(n+\frac{1}{2}% )^{2}}\cos\left((2n+1)z\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.27.2 Permalink: http://dlmf.nist.gov/20.2.E2 Encodings: TeX, pMML, png See also: Annotations for §20.2(i), §20.2 and Ch.20
20.2.3	$\displaystyle\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(z,q\right)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos% \left(2nz\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.27.3 Referenced by: §20.10(i), §20.13, §20.14, §27.13(iv) Permalink: http://dlmf.nist.gov/20.2.E3 Encodings: TeX, pMML, png See also: Annotations for §20.2(i), §20.2 and Ch.20
20.2.4	$\displaystyle\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(z,q\right)=1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2% }}\cos\left(2nz\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.27.4 Referenced by: §20.13, §20.2(i) Permalink: http://dlmf.nist.gov/20.2.E4 Encodings: TeX, pMML, png See also: Annotations for §20.2(i), §20.2 and Ch.20

Corresponding expansions for $\theta_{j}'\left(z\middle|\tau\right)$ , $j=1,2,3,4$ , can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$ .

§20.2(ii) Periodicity and Quasi-Periodicity

ⓘ

Keywords:: fundamental parallelogram, lattice points, periodicity, quasi-periodicity, theta functions
Notes:: See Whittaker and Watson (1927, p. 463) and Lawden (1989, pp. 5, 6).
Permalink:: http://dlmf.nist.gov/20.2.ii
See also:: Annotations for §20.2 and Ch.20

For fixed $\tau$ , each $\theta_{j}\left(z\middle|\tau\right)$ is an entire function of $z$ with period $2\pi$ ; $\theta_{1}\left(z\middle|\tau\right)$ is odd in $z$ and the others are even. For fixed $z$ , each of $\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}$ , $\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}$ , $\theta_{3}\left(z\middle|\tau\right)$ , and $\theta_{4}\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\Im\tau>0$ , with a natural boundary $\Im\tau=0$ , and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ with a natural boundary $\left|q\right|=1$ .

The four points $(0,\pi,\pi+\tau\pi,\tau\pi)$ are the vertices of the fundamental parallelogram in the $z$ -plane; see Figure 20.2.1. The points

20.2.5		$z_{m,n}=(m+n\tau)\pi,$
$m,n\in\mathbb{Z}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathbb{Z}$ : set of all integers, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter A&S Ref: 16.33.1 (in different notation) Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.2.E5 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20

are the lattice points. The theta functions are quasi-periodic on the lattice:

20.2.6	$\displaystyle\theta_{1}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=(-1)^{m+n}q^{-n^{2}}e^{-2inz}\theta_{1}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.33.2 (in different notation) Referenced by: §23.6(i) Permalink: http://dlmf.nist.gov/20.2.E6 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20
20.2.7	$\displaystyle\theta_{2}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=(-1)^{m}q^{-n^{2}}e^{-2inz}\theta_{2}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.33.3 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E7 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20
20.2.8	$\displaystyle\theta_{3}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=q^{-n^{2}}e^{-2inz}\theta_{3}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.33.4 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E8 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20
20.2.9	$\displaystyle\theta_{4}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=(-1)^{n}q^{-n^{2}}e^{-2inz}\theta_{4}\left(z\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome Permalink: http://dlmf.nist.gov/20.2.E9 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20

See accompanying text — Figure 20.2.1: $z$ -plane. Fundamental parallelogram. Left-hand diagram is the rectangular case ( $\tau$ purely imaginary); right-hand diagram is the general case. $\bullet$ zeros of $\theta_{1}\left(z\middle|\tau\right)$ , $\blacksquare$ zeros of $\theta_{2}\left(z\middle|\tau\right)$ , $\blacktriangle$ zeros of $\theta_{3}\left(z\middle|\tau\right)$ , $\blacklozenge$ zeros of $\theta_{4}\left(z\middle|\tau\right)$ . Magnify

§20.2(iii) Translation of the Argument by Half-Periods

ⓘ

Keywords:: theta functions, translation by half-periods
Notes:: See Whittaker and Watson (1927, p. 464) and Lawden (1989, pp. 5, 6).
Referenced by:: §21.3(ii), Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/20.2.iii
See also:: Annotations for §20.2 and Ch.20

With

20.2.10		$M\equiv M(z\|\tau)=e^{iz+(i\pi\tau/4)},$
ⓘ Defines: $M$ (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\equiv$ : equals by definition, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter A&S Ref: 16.33.1 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E10 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20

20.2.11	$\displaystyle\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=-\theta_{2}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=-iM\theta_% {4}\left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=-iM\theta_{3}\left(z+\tfrac{% 1}{2}\pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex, $\tau$ : lattice parameter and $M$ A&S Ref: 16.33.2 (in different notation) Referenced by: §20.7(iv) Permalink: http://dlmf.nist.gov/20.2.E11 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
20.2.12	$\displaystyle\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{1}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=M\theta_{3}% \left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=M\theta_{4}\left(z+\tfrac{1}{2}% \pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $M$ A&S Ref: 16.33.3 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E12 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
20.2.13	$\displaystyle\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=M\theta_{2}% \left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=M\theta_{1}\left(z+\tfrac{1}{2}% \pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $M$ A&S Ref: 16.33.4 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E13 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
20.2.14	$\displaystyle\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=-iM\theta_{% 1}\left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=iM\theta_{2}\left(z+\tfrac{1}% {2}\pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex, $\tau$ : lattice parameter and $M$ Referenced by: §20.7(iv) Permalink: http://dlmf.nist.gov/20.2.E14 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20

§20.2(iv) $z$ -Zeros

ⓘ

Keywords:: theta functions, zeros
Notes:: See Whittaker and Watson (1927, p. 465) and Lawden (1989, p. 6).
Permalink:: http://dlmf.nist.gov/20.2.iv
See also:: Annotations for §20.2 and Ch.20

For $m,n\in\mathbb{Z}$ , the $z$ -zeros of $\theta_{j}\left(z\middle|\tau\right)$ , $j=1,2,3,4$ , are $(m+n\tau)\pi$ , $(m+\tfrac{1}{2}+n\tau)\pi$ , $(m+\tfrac{1}{2}+(n+\tfrac{1}{2})\tau)\pi$ , $(m+(n+\tfrac{1}{2})\tau)\pi$ respectively.

20.1 Special Notation 20.3 Graphics

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

20.2.6	$\displaystyle\theta_{1}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=(-1)^{m+n}q^{-n^{2}}e^{-2inz}\theta_{1}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.33.2 (in different notation) Referenced by: §23.6(i) Permalink: http://dlmf.nist.gov/20.2.E6 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20
20.2.7	$\displaystyle\theta_{2}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=(-1)^{m}q^{-n^{2}}e^{-2inz}\theta_{2}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.33.3 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E7 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20
20.2.8	$\displaystyle\theta_{3}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=q^{-n^{2}}e^{-2inz}\theta_{3}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome A&S Ref: 16.33.4 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E8 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20
20.2.9	$\displaystyle\theta_{4}\left(z+(m+n\tau)\pi\middle\|\tau\right)$	$\displaystyle=(-1)^{n}q^{-n^{2}}e^{-2inz}\theta_{4}\left(z\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome Permalink: http://dlmf.nist.gov/20.2.E9 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20

20.2.11	$\displaystyle\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=-\theta_{2}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=-iM\theta_% {4}\left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=-iM\theta_{3}\left(z+\tfrac{% 1}{2}\pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex, $\tau$ : lattice parameter and $M$ A&S Ref: 16.33.2 (in different notation) Referenced by: §20.7(iv) Permalink: http://dlmf.nist.gov/20.2.E11 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
20.2.12	$\displaystyle\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{1}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=M\theta_{3}% \left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=M\theta_{4}\left(z+\tfrac{1}{2}% \pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $M$ A&S Ref: 16.33.3 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E12 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
20.2.13	$\displaystyle\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=M\theta_{2}% \left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=M\theta_{1}\left(z+\tfrac{1}{2}% \pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $M$ A&S Ref: 16.33.4 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E13 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
20.2.14	$\displaystyle\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(z+\tfrac{1}{2}\pi\middle\|\tau\right)=-iM\theta_{% 1}\left(z+\tfrac{1}{2}\pi\tau\middle\|\tau\right)=iM\theta_{2}\left(z+\tfrac{1}% {2}\pi+\tfrac{1}{2}\pi\tau\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex, $\tau$ : lattice parameter and $M$ Referenced by: §20.7(iv) Permalink: http://dlmf.nist.gov/20.2.E14 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20

§20.2 Definitions and Periodic Properties

Contents

§20.2(i) Fourier Series

§20.2(ii) Periodicity and Quasi-Periodicity

§20.2(iii) Translation of the Argument by Half-Periods

§20.2(iv) $z$ -Zeros

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

§20.2 Definitions and Periodic Properties

Contents

§20.2(i) Fourier Series

§20.2(ii) Periodicity and Quasi-Periodicity

§20.2(iii) Translation of the Argument by Half-Periods

§20.2(iv) z-Zeros

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

§20.2(iv) $z$ -Zeros