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

## Functions of a hypercomplex variable

### Polynomials, Rationals, Exponentials, and Logarithms

#### 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])

#### 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 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])

#### 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])

#### 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