Arithmetic |
Hypercomplex components |
Polynomials, Rationals, Exponentials, and Logarithms |
Trigonometric functions and inverses |
Hyperbolic functions and inverses |

Sinc analogues |
Logistic functions, Gudermannian, and inverses |
Algebraic curves |
Bessel-type functions |
Integer functions |

Factorials and binomial coefficient |
Productlogarithms, hyperoperators, and relatives |
Physical functions |
L^{p} trigonometric functions |
Exponential integral and relatives |

Elliptic integrals and relatives |
Gamma functions and relatives |
Error functions and relatives |
Zeta functions, Polylogarithms, and relatives |
Modular forms and relatives |

## Functions of a hypercomplex variable

### Polynomials, Rationals, Exponentials, and Logarithms

#### Square

- Complex square (complex map v1 / complex map v2/ 3D plot / sliders / domain colouring / contour plot / Mandelbrot)
- is elementary
- is complex analytic
- has a zero of order 2 at
- has a pole of order 2 at
- is conformal except at
- is entire

- Dual square ([ dual map] / 3D plot / sliders)
- Split-complex square ([ split-complex map] / 3D plot / sliders)
- Quaternionic square (vectors / sliders / 3D rotations)
- Octonionic square (vectors / sliders)
- Split-quaternionic square (vectors / [ sliders])
- Ternionic square (vectors / [ sliders])
- Split-octonionic square (vectors / [ sliders])
- Tessarine (bicomplex) square (vectors / [ sliders])

#### Cube

- Complex cube (complex map v1 / complex map v2 / 3D plot / domain colouring / contour plot / Mandelbrot)
- is elementary
- is complex analytic
- has a zero of order 3 at
- has a pole of order 3 at
- is conformal except at
- is entire

- Dual cube ([ dual map] / 3D plot)
- Split-complex cube ([ split-complex map] / 3D plot)
- Quaternionic cube (vectors / sliders / 3D rotations)
- Octonionic cube (vectors / sliders)

#### Tesseract

- Complex tesseract (complex map v1 / complex map v2 / 3D plot / domain colouring / contour plot / Mandelbrot)
- is elementary
- is complex analytic
- has a zero of order 4 at
- has a pole of order 4 at
- is conformal except at
- is entire

- Dual tesseract ([ dual map] / [ 3D plot])
- Split-complex tesseract ([ split-complex map] / [ 3D plot])
- Quaternionic tesseract ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic tesseract ([ vectors] / [ sliders])

#### Constant functions

#### Linear functions

- Complex linear functions (complex map / polygons in the Euclidean plane / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Dual linear functions ([ dual map] / [ 3D plot] / [ sliders])
- Split-complex linear functions ([ split-complex map] / [ 3D plot] / [ sliders])
- Quaternionic linear functions ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic linear functions ([ vectors] / [ sliders])

#### Linear fractions

- Complex linear fractions ([ complex map] / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Complex Riemann sphere rotating MΓΆbius transformations (complex map / [ polygons in a stereographic projection of a sphere] / stereographic projection of Earth's night sky / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Complex unit-disk preserving MΓΆbius transformations (complex map / polygons in the PoincarΓ© disc model of the hyperbolic plane / pseudogons in the PoincarΓ© disc model of the hyperbolic plane / 3D plot / [ domain colouring] / [ contour plot])
- Complex upper half-plane preserving MΓΆbius transformations ([ complex map] / polygons in the PoincarΓ© half-plane model of the hyperbolic plane / [ 3D plot] / [ domain colouring] / [ contour plot])
- Dual linear fractions ([ dual map] / [ 3D plot] / [ sliders])
- Split-complex linear fractions ([ split-complex map] / [ 3D plot] / [ sliders])
- Quaternionic linear fractions ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic linear fractions ([ vectors] / [ sliders])

#### Quadratic functions

- Complex quadratic functions (complex map / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Dual quadratic functions ([ dual map] / [ 3D plot] / [ sliders])
- Split-complex quadratic functions ([ split-complex map] / [ 3D plot] / [ sliders])
- Quaternionic quadratic functions ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic quadratic functions ([ vectors] / [ sliders])

#### Cubic functions

- Complex cubic functions (complex map / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Dual cubic functions ([ dual map] / [ 3D plot] / [ sliders])
- Split-complex cubic functions ([ split-complex map] / [ 3D plot] / [ sliders])
- Quaternionic cubic functions ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic cubic functions ([ vectors] / [ sliders])

#### Quartic functions

- Complex quartic functions ([ complex map] / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Dual quartic functions ([ dual map] / [ 3D plot] / [ sliders])
- Split-complex quartic functions ([ split-complex map] / [ 3D plot] / [ sliders])
- Quaternionic quartic functions ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic quartic functions ([ vectors] / [ sliders])

#### Exponentiation

- Complex powers (complex map of principal branch v1 / complex map v2 / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot / Mandelbrot)
- is elementary
- is analytic on
- has a zero of order at and a zero of order at
- has pole of order at and a pole of order at
- is conformal except at if

- Complex exponentials (complex map / 3D plot / domain colouring / contour plot)
- Complex double exponentials (complex map / 3D plot / domain colouring / contour plot)
- Dual powers (dual map / 3D plot)
- Dual exponentials (dual map / 3D plot)
- Dual double exponentials ([ complex map] / [ 3D plot])
- Split-complex powers (split-complex map / 3D plot)
- Split-complex exponentials (split-complex map / 3D plot)
- Split-complex double exponentials ([ complex map] / [ 3D plot])
- Quaternionic powers ([ vectors] / [ sliders] / [ 3D rotations])
- Quaternionic exponentials ([ vectors] / [ sliders] / [ 3D rotations])
- Quaternionic double exponentials ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic powers ([ vectors] / [ sliders])
- Octonionic exponentials ([ vectors] / [ sliders])
- Octonionic double exponentials ([ vectors] / [ sliders])

#### Square root

- Complex square root (complex map of principal branch v1 / complex map of principal branch v2 / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on
- has a zero of order at
- has a pole of order at
- is conformal except at
- is multivalued
- has a branch point at

- Dual square root ([ dual map of principal branch] / 3D plot of principal value)
- is extremely multivalued at

- Split-complex square root ([ split-complex map of principal branch] / 3D plot of principal value)
- Quaternionic square root (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic square root (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

#### Cube root

- Complex cube root (complex map of principal branch v1 / complex map of principal branch v2 / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on
- has a zero of order at
- has a pole of order at
- is conformal except at
- is multivalued
- has a branch point at

- Dual cube root ([ dual map] / 3D plot)
- is extremely multivalued at

- Split-complex cube root ([ split-complex map] / 3D plot)
- Quaternionic cube root (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic cube root (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

#### Tesseract root

- Complex tesseract root (complex map of principal branch v1 / complex map of principal branch v2 / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on
- has a zero of order at
- has a pole of order at
- is conformal except at
- is multivalued
- has a branch point at

- Dual tesseract root ([ dual map of principal value] / 3D plot of principal value)
- is extremely multivalued at

- Split-complex tesseract root ([ split-complex map of principal value] / 3D plot of principal value)
- Quaternionic tesseract root (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic tesseract root (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

#### Roots

- Complex th roots (complex map of principal branch v1 / complex map of principal branch v2 / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on
- has a zero of order at and a zero of order at
- has a pole of order at and a pole of order at
- is conformal except at if

- Complex (complex map of principal branch / complex maps of branches / vectors of values / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- Dual th roots ([ dual map] / [ 3D plot])
- Split-complex th roots ([ split-complex map] / [ 3D plot])

#### Exponential

- Complex exponential (complex map / 3D plot / domain colouring / contour plot / Mandelbrot / iterates)
- is elementary
- is complex analytic
- has an essential singularity at
- is conformal
- is entire
- forms an algebraically closed exponential field
- Complex reciprocal of exponential () (complex map / 3D plot / [ domain colouring] / [ contour plot])
- Complex exponential of reciprocal () (complex map / 3D plot / [ domain colouring] / [ contour plot])
- Complex natural logarithm of exponential ()
- Complex exponential of natural logarithm ()
- Complex inverse of exponential ()
- Complex derivative of exponential ()
- Complex integral of exponential () (complex map / 3D plot / [ domain colouring] / [ contour plot])
- Complex exponential of exponential () (complex map / 3D plot / [ domain colouring] / [ contour plot])

- Dual exponential (dual map / 3D plot)
- Split-complex exponential (split-complex map / 3D plot)
- Quaternionic exponential (vectors / sliders / 3D rotations)
- Octonionic exponential (vectors / sliders)
- Split-quaternionic exponential (vectors / [ sliders])
- Ternionic exponential (vectors / [ sliders])
- Split-octonionic exponential (vectors / [ sliders])
- Tessarine (bicomplex) exponential (vectors / [ sliders])

#### Natural logarithm

- Complex natural logarithm (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot / Mandelbrot)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and
- Complex reciprocal of natural logarithm () (complex map of principal branch / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- Complex natural logarithm of reciprocal () (complex map of principal branch / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- Complex natural logarithm of natural logarithm () (complex map of principal branch / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- Complex inverse of natural logarithm ()
- Complex derivative of natural logarithm ()
- Complex integral of natural logarithm () (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])

- Dual natural logarithm (dual map / 3D plot)
- Split-complex natural logarithm (split-complex map / 3D plot)
- Quaternionic natural logarithm (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic natural logarithm (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

- Split-quaternionic natural logarithm (vectors of principal value / [ sliders of principal value])
- is extremely multivalued for elements of

- Ternionic natural logarithm (vectors of principal value / [ sliders of principal value])
- Split-octonionic natural logarithm (vectors of principal value / [ sliders of principal value])
- is extremely multivalued for elements of

- Tessarine (bicomplex) natural logarithm (vectors of principal value / [ sliders of principal value])
- is extremely multivalued for elements of

#### Binary logarithm

- Complex binary logarithm (complex map of principal branch / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and

- Dual binary logarithm (dual map / 3D plot)
- Split-complex binary logarithm (split-complex map / 3D plot)
- Quaternionic binary logarithm ([ vectors] / [ sliders] / [ 3D rotations])
- is extremely multivalued for elements of

- Octonionic binary logarithm ([ vectors] / [ sliders])
- is extremely multivalued for elements of

#### Decimal logarithm

- Complex decimal logarithm (complex map of principal branch / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and

- Dual decimal logarithm (dual map / 3D plot)
- Split-complex decimal logarithm (split-complex map / 3D plot)
- Quaternionic decimal logarithm (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic decimal logarithm (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

#### Quater-imaginary logarithm

- Complex quater-imaginary logarithm (complex map of principal branch / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and

- Quaternionic quater-imaginary logarithm ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic quater-imaginary logarithm ([ vectors] / [ sliders])

#### Logarithms

- Complex logarithms (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and

- Dual logarithms ([ dual map] / [ 3D plot])
- Split-complex logarithms ([ split-complex map] / [ 3D plot])
- Quaternionic logarithms ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic logarithms ([ vectors] / [ sliders])

#### Exponential minus one

- Complex exponential minus one (complex map / 3D plot / domain colouring / contour plot)
- is elementary
- is complex analytic
- has a simple zero at
- has an essential singularity at
- is conformal
- is entire

- Dual exponential minus one (dual map / 3D plot)
- Split-complex exponential minus one (split-complex map / 3D plot)
- Quaternionic exponential minus one (vectors / sliders / 3D rotations)
- Octonionic exponential minus one (vectors / sliders)
- Split-quaternionic exponential minus one (vectors / [ sliders])
- Ternionic exponential minus one (vectors / [ sliders])
- Split-octonionic exponential minus one (vectors / [ sliders])
- Tessarine (bicomplex) exponential minus one (vectors / [ sliders])

#### Natural logarithm one plus

- Complex natural logarithm one plus (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface / domain colouring / contour plot)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and

- Dual natural logarithm one plus (dual map / 3D plot)
- Split-complex natural logarithm one plus (split-complex map / 3D plot)
- Quaternionic natural logarithm one plus (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic natural logarithm one plus (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

- Split-quaternionic natural logarithm one plus ([ vectors of principal value] / [ sliders of principal value])
- is extremely multivalued for elements of

- Ternionic natural logarithm one plus ([ vectors of principal value] / [ sliders of principal value])
- Split-octonionic natural logarithm one plus ([ vectors of principal value] / [ sliders of principal value])
- is extremely multivalued for elements of

- Tessarine (bicomplex) natural logarithm one plus ([ vectors of principal value] / [ sliders of principal value])
- is extremely multivalued for elements of

#### Periexp

- Complex periexp (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- is elementary
- is analytic on any open set on its domain
- has a simple pole at
- is conformal
- is multivalued
- has a branch point at 0

- Quaternionic periexp ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic periexp ([ vectors] / [ sliders])

#### Gaussian functions

- Complex Gaussian functions (complex map / 3D plot / [ sliders] / [ domain colouring] / [ contour plot])
- Dual Gaussian functions ([ dual map] / [ 3D plot] / [ sliders])
- Split-complex Gaussian functions ([ split-complex map] / [ 3D plot] / [ sliders])
- Quaternionic Gaussian functions ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic Gaussian functions ([ vectors] / [ sliders])

#### Lorentzian functions

#### Trappmann function

- Complex Trappmann function (complex map / 3D plot / [ domain colouring] / [ contour plot])
- is elementary
- is complex analytic
- has simple zeros at all
- is conformal except at elements of
- is entire

- Dual Trappmann function (dual map / 3D plot)
- Split-complex Trappmann function (split-complex map / 3D plot)
- Quaternionic Trappmann function (vectors / sliders / 3D rotations)
- Octonionic Trappmann function (vectors / sliders)

#### Zex

- Complex zex (complex map / 3D plot / [ domain colouring] / [ contour plot])
- is elementary
- is complex analytic
- has a simple zero at
- is conformal except at
- is entire

- Dual zex (dual map / 3D plot)
- Split-complex zex (split-complex map / 3D plot)
- Quaternionic zex (vectors / sliders / 3D rotations)
- Octonionic zex (vectors / sliders)

#### Zog

- Complex zog (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- is elementary
- is complex analytic
- is conformal except at and
- is multivalued
- has a branch point at

- Dual zog ([ dual map] / [ 3D plot])
- Split-complex zog ([ split-complex map] / [ 3D plot])
- Quaternionic zog ([ vectors] / [ sliders] / [ 3D rotations])
- is extremely multivalued for elements of

- Octonionic zog ([ vectors] / [ sliders])
- is extremely multivalued for elements of

#### Keller function

- Complex Keller function (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- Dual Keller function ([ dual map] / [ 3D plot])
- Split-complex Keller function ([ split-complex map] / [ 3D plot])
- Quaternionic Keller function ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic Keller function ([ vectors] / [ sliders])

#### Inverse Keller function

- Complex inverse Keller function (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- Dual inverse Keller function ([ dual map] / [ 3D plot])
- Split-complex inverse Keller function ([ split-complex map] / [ 3D plot])
- Quaternionic inverse Keller function ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic inverse Keller function ([ vectors] / [ sliders])

#### Shoka function

- Complex Shoka function (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- Dual Shoka function ([ dual map] / [ 3D plot])
- Split-complex Shoka function ([ split-complex map] / [ 3D plot])
- Quaternionic Shoka function ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic Shoka function ([ vectors] / [ sliders])

#### Inverse Shoka function

- Complex inverse Shoka function (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- Dual inverse Shoka function ([ dual map] / [ 3D plot])
- Split-complex inverse Shoka function ([ split-complex map] / [ 3D plot])
- Quaternionic inverse Shoka function ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic inverse Shoka function ([ vectors] / [ sliders])

#### Inverse Tania function

- Complex inverse Tania function (complex map of principal branch / 3D plot of principal branch / Riemann surface / [ domain colouring] / [ contour plot])
- is elementary
- is analytic on any open set on its domain
- is conformal except at
- is multivalued
- has a branch point at

- Dual inverse Tania function ([ dual map] / [ 3D plot])
- Split-complex inverse Tania function ([ split-complex map] / [ 3D plot])
- Quaternionic inverse Tania function ([ vectors] / [ sliders] / [ 3D rotations])
- is extremely multivalued for elements of

- Octonionic inverse Tania function ([ vectors] / [ sliders])
- is extremely multivalued for elements of

#### Anka function

- Complex Anka function (complex map / 3D plot / [ domain colouring] / [ contour plot])
- is elementary
- is complex analytic
- has a simple zero at
- is conformal except at
- is entire

- Dual Anka function (dual map / 3D plot)
- Split-complex Anka function (split-complex map / 3D plot)
- Quaternionic Anka function (vectors / sliders / 3D rotations)
- Octonionic Anka function (vectors / sliders)