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

#### Constant zero function

- Complex constant zero function (complex map / 3D plot / sliders / domain colouring)
- is elementary
- is complex analytic
- is entire

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

#### Constant one function

- Complex constant one function (complex map / 3D plot / sliders / domain colouring)
- is elementary
- is complex analytic
- is entire

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

#### Identity function

- Complex identity function (complex map / 3D plot / sliders / domain colouring / contour plot)
- is an involution
- is elementary
- is complex analytic
- has a simple zero at
- has a simple pole at
- is conformal
- is entire

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

### Arithmetic

#### Addition

- Complex addition (complex map / 3D plot / [ sliders] / domain colouring / contour plot)
- forms an abelian group with identity element
- is elementary
- is complex analytic
- has a simple zero at
- has a simple pole at
- is conformal
- is entire

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

#### Multiplication

**Complex multiplication**(complex map / 3D plot / [ sliders] / domain colouring / contour plot)- forms an abelian group with identity element
- is elementary
- is complex analytic
- has a simple zero at
- has a simple pole at
- is conformal except if
- is entire

**Dual multiplication**(dual map / 3D plot / [ sliders])**Split-complex multiplication**(split-complex map / (Spacetime globes: Position-time spacetime globe / Time-position spacetime globe) / 3D plot / [ sliders])**Quaternionic multiplication**(vectors / [ sliders] / 3D rotations)**Octonionic multiplication**(vectors, showcasing noncommmutativity / vectors, showcasing nonassociativity / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])**Split-quaternionic multiplication**(vectors / [ sliders])**Ternionic multiplication**(vectors / [ sliders])**Split-octonionic multiplication**(vectors, showcasing noncommmutativity / vectors, showcasing nonassociativity / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])**Quintonionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])**Sextonionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])**Tessarine (bicomplex) multiplication**(vectors / [ sliders])**Octarine (biquaternionic) multiplication**([ vectors, showcasing noncommmutativity] / [ sliders, showcasing noncommutativity])**Hexadecarine (bioctonionic) multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])**Sedenionic multiplication**(vectors, showcasing noncommmutativity / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])**Trigintaduonionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])**Tricomplex multiplication**([ vectors] / [ sliders])**Tetracomplex multiplication**([ vectors] / [ sliders])**Pentacomplex multiplication**([ vectors] / [ sliders])**Split-biquaternionic multiplication**([ vectors, showcasing noncommmutativity] / [ sliders, showcasing noncommutativity])**Dual complex multiplication**([ vectors] / [ sliders])**Dual quaternion multiplication**([ vectors] / [ sliders])**Parabolic quaternionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])**Hyperbolic quaternionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])**Parabolic octonionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])**Hyperbolic octonionic multiplication**([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])

#### Negation

- Complex negation (complex map / 3D plot / sliders / domain colouring / contour plot)
- is an involution
- is elementary
- is complex analytic
- has a simple zero at
- has a simple pole at
- is conformal
- is entire
- forms an algebraically closed ring

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

#### Reciprocal

- Complex reciprocal (complex map v1 / complex map v2 / 3D plot / sliders / domain colouring / contour plot)
- is an involution
- is elementary
- is analytic on
- is meromorphic
- has a simple zero at
- has a simple pole at
- is conformal
- forms an algebraically closed field

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