ℍ𝕪𝕡𝕖𝕣𝕔𝕠𝕞𝕡𝕝𝕖𝕩 𝕤𝕥𝕦𝕗𝕗/𝔽𝕦𝕟𝕔𝕥𝕚𝕠𝕟𝕤 𝕠𝕗 𝕒 ℍ𝕪𝕡𝕖𝕣𝕔𝕠𝕞𝕡𝕝𝕖𝕩 𝕍𝕒𝕣𝕚𝕒𝕓𝕝𝕖/ℍ𝕪𝕡𝕖𝕣𝕔𝕠𝕞𝕡𝕝𝕖𝕩 𝕔𝕠𝕞𝕡𝕠𝕟𝕖𝕟𝕥𝕤

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

Complex $z^{*}$ domain colouring

• Complex conjugation (complex map / 3D plot / sliders / domain colouring / contour plot)
  • ${}^{*}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${}^{*}$ 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])

Real part

Complex $\Re (z)$ domain colouring

3D plot of $\Re (z)$

Imaginary part

Complex $\Im (z)$ domain colouring

3D plot of $\Im (z)$

Absolute value

For an arbitrary element $z$ in an arbitrary hypercomplex number system $H$, $|z|:={\sqrt {zz^{*}}}$, the modulus of $z$, where ${}^{*}$ is an involution called conjugation. The properties of conjugation dictate the properties of the modulus.

$|z|$ is an absolute value if it is a real-valued function that satisfies

• Non-negativity ($0\leq |z|$)
• Positive-definiteness ($|z|=0\iff z=0$)
• Multiplicativity ($|zw|=|z|\cdot |w|$)
• and Subadditivity ($|z+w|\leq |z|+|w|$)

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Complex $|z|$ domain colouring

Complex $|z|^{2}$ domain colouring

3D plot of $|z|$

• Complex absolute value (complex map / 3D plot / sliders / domain colouring)
  • $||:{\mathbb {C} }\to {\mathbb {R} }_{0\leq }$
  • $||$ is non-analytic
• Complex absolute value squared (complex map / 3D plot / sliders / domain colouring)
  • $||^{2}:{\mathbb {C} }\to {\mathbb {R} }_{0\leq }$
  • $||^{2}$ 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 $\{z\in {\mathbb {P} }:|z|^{2}\in {\mathbb {R} }_{<0}\}$
• Split-quaternionic modulus squared (vectors / [ sliders])
• Split-octonionic modulus (vectors / [ sliders])
  • $||$ is extremely multivalued for elements of $\{z\in {\mathbb {P} }:|z|^{2}\in {\mathbb {R} }_{<0}\}$
• 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 $\{z\in {\mathbb {P} }:|z|^{2}\in {\mathbb {R} }_{<0}\}$
• Tessarine (bicomplex) bicomplex modulus squared (vectors / [ sliders])

Signum

Complex ${\rm {sign}}(z)$ domain colouring

• Complex signum (complex map / 3D plot / sliders / domain colouring)
  • ${\rm {sign}}:{\mathbb {C} }\to \{z\in {\mathbb {C} }:|z|=1\}\cup \{0\}$
  • ${\rm {sign}}$ is non-analytic
• Dual signum (dual map / 3D plot / sliders)
• Split-complex signum (split-complex map / 3D plot / sliders)
• Quaternionic signum (vectors / sliders / 3D rotations)
• Octonionic signum (vectors / sliders)

Argument

Complex ${\rm {arg}}(z)$ domain colouring

3D plot of ${\rm {arg}}(z)$

• Complex argument (complex map of principal branch / 3D plot of principal value / sliders / domain colouring)
  • ${\rm {arg}}:{\mathbb {C} }\to {\mathbb {R} }$
  • ${\rm {arg}}$ is non-analytic
  • ${\rm {arg}}$ is multivalued
• Dual argument (dual map / 3D plot / sliders)
• Split-complex argument (split-complex map / 3D plot / sliders)
• Quaternionic argument (vectors / sliders / 3D rotations)
• Octonionic argument (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])

References

Graphs

• 3D Graph using Parametric Lines
• {mod} 3D Cube using Parametric Lines