Illustrations of Eisenstein series

Image: X-ray of

E_{2}\!\left(\tau\right)

on

\tau \in \left[-1, 1\right] + \left[0, 2\right] i

with

\mathcal{F}

highlighted

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("7e1850"),
    Image(Description("X-ray of", EisensteinE(2, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e2")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

dac0bb

Image: X-ray of

E_{4}\!\left(\tau\right)

on

\tau \in \left[-1, 1\right] + \left[0, 2\right] i

with

\mathcal{F}

highlighted

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("dac0bb"),
    Image(Description("X-ray of", EisensteinE(4, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e4")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

54951a

Image: X-ray of

E_{6}\!\left(\tau\right)

on

\tau \in \left[-1, 1\right] + \left[0, 2\right] i

with

\mathcal{F}

highlighted

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("54951a"),
    Image(Description("X-ray of", EisensteinE(6, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e6")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

356d7a

Image: X-ray of

E_{8}\!\left(\tau\right)

on

\tau \in \left[-1, 1\right] + \left[0, 2\right] i

with

\mathcal{F}

highlighted

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("356d7a"),
    Image(Description("X-ray of", EisensteinE(8, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e8")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

626026

Image: X-ray of

E_{10}\!\left(\tau\right)

on

\tau \in \left[-1, 1\right] + \left[0, 2\right] i

with

\mathcal{F}

highlighted

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("626026"),
    Image(Description("X-ray of", EisensteinE(10, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e10")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

a14cfc

Image: X-ray of

E_{12}\!\left(\tau\right)

on

\tau \in \left[-1, 1\right] + \left[0, 2\right] i

with

\mathcal{F}

highlighted

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("a14cfc"),
    Image(Description("X-ray of", EisensteinE(12, tau), "on", Element(tau, Add(ClosedInterval(-1, 1), Mul(ClosedInterval(0, 2), ConstI))), "with", ModularGroupFundamentalDomain, "highlighted"), ImageSource("xray_eisenstein_e12")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

X-ray plots