## Functions of a hypercomplex variable

### Hypercomplex components

#### Conjugation

• Complex conjugation (complex map / 3D plot / sliders / domain colouring / contour plot)
• • is an involution
• is non-analytic
• is isogonal
• Dual conjugation (dual map / 3D plot / sliders)
• Split-complex conjugation (split-complex map / 3D plot / sliders)
• Quaternionic conjugation (vectors / sliders / 3D rotations)
• Octonionic conjugation (vectors / sliders)
• Split-quaternionic conjugation (vectors / [ sliders])
• Split-octonionic conjugation (vectors / [ sliders])

#### Absolute value

For an arbitrary element in an arbitrary hypercomplex number system , , the modulus of , where is an involution called conjugation. The properties of conjugation dictate the properties of the modulus. is an absolute value if it is a real-valued function that satisfies

• Non-negativity ( )
• Positive-definiteness ( )
• Multiplicativity ( )
• and Subadditivity ( )

-

• Complex absolute value (complex map / 3D plot / sliders / domain colouring)
• • is non-analytic
• Complex absolute value squared (complex map / 3D plot / sliders / domain colouring)
• • is non-analytic
• Dual modulus (dual map / 3D plot / sliders)
• Dual modulus squared (dual map / 3D plot / sliders)
• Split-complex modulus squared (split-complex map / 3D plot / sliders)
• Split-complex tessarine modulus of the tessarine modulus (split-complex map / 3D plot / sliders)
• Quaternionic absolute value (vectors / sliders / 3D rotations)
• Quaternionic absolute value squared (vectors / sliders / 3D rotations)
• Octonionic absolute value (vectors / sliders)
• Octonionic absolute value squared (vectors / sliders)
• Split-quaternionic modulus (vectors / [ sliders])
• is extremely multivalued for elements of • Split-quaternionic modulus squared (vectors / [ sliders])
• Split-octonionic modulus (vectors / [ sliders])
• is extremely multivalued for elements of • Split-octonionic modulus squared (vectors / [ sliders])
• Tessarine (bicomplex) tessarine modulus (vectors / [ sliders])
• Tessarine (bicomplex) tessarine modulus squared (vectors / [ sliders])
• Tessarine (bicomplex) bicomplex modulus (vectors / [ sliders])
• is extremely multivalued for elements of • Tessarine (bicomplex) bicomplex modulus squared (vectors / [ sliders])

#### Distance function

• Complex Euclidean metric ([ complex map] / [ 3D plot])
• Complex Euclidean metric squared ([ complex map] / [ 3D plot])
• Dual spacetime interval ([ dual map] / [ 3D plot])
• Split-complex spacetime interval ([ split-complex map] / [ 3D plot])

#### Commutator

• Quaternionic commutator (vectors / [ sliders] / [ 3D rotations])
• Octonionic commutator ([ vectors] / [ sliders] / [ 3D rotations])

#### Anticommutator

• Quaternionic anticommutator (vectors / [ sliders] / [ 3D rotations])
• Octonionic anticommutator ([ vectors] / [ sliders] / [ 3D rotations])