Verse and Dimensions Wikia
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Verse and Dimensions Wikia


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
Lp 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

Hypercomplex components

Conjugation

Real part

Imaginary part

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 ()

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Signum

Argument

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])

Associator

Antiassociator

References

Graphs

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