ℍ𝕪𝕡𝕖𝕣𝕔𝕠𝕞𝕡𝕝𝕖𝕩 𝕤𝕥𝕦𝕗𝕗/𝔽𝕦𝕟𝕔𝕥𝕚𝕠𝕟𝕤 𝕠𝕗 𝕒 ℍ𝕪𝕡𝕖𝕣𝕔𝕠𝕞𝕡𝕝𝕖𝕩 𝕍𝕒𝕣𝕚𝕒𝕓𝕝𝕖/𝕋𝕣𝕚𝕘𝕠𝕟𝕠𝕞𝕖𝕥𝕣𝕚𝕔 𝕗𝕦𝕟𝕔𝕥𝕚𝕠𝕟𝕤 𝕒𝕟𝕕 𝕚𝕟𝕧𝕖𝕣𝕤𝕖𝕤

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

Trigonometric functions and inverses

Sine

Complex $\sin(z)$ contour plot

Complex $\sin(z)$ domain colouring

Dual $\sin(z)$ contour plot

Dual $\sin(z)$ domain colouring

Split-complex $\sin(z)$ contour plot

Split-complex $\sin(z)$ domain colouring

Complex $\sin(z^{-1})$ contour plot

Complex $\sin(z^{-1})$ domain colouring

Complex $\log(\sin(z))$ contour plot

Complex $\log(\sin(z))$ domain colouring

Complex $\sin(\log(z))$ contour plot. Branch cut is $(-\infty ,0]$.

Complex $\sin(\log(z))$ domain colouring. Branch cut is $(-\infty ,0]$.

Complex $\sin(\sin(z))$ contour plot

Complex $\sin(\sin(z))$ domain colouring

$\sin(z):=\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{(2n+1)!}}$

• Complex sine (complex map / 3D plot / Mandelbrot / iterates)
  • $\sin :{\mathbb {C} }\to {\mathbb {C} }$
  • $\sin$ is elementary
  • $\sin$ is complex analytic
  • $\sin$ has simple zeros at all $n\pi :n\in {\mathbb {Z} }$
  • $\sin$ has an essential singularity at $\infty$
  • $\sin$ is conformal except at elements of $\{\pi n+{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}$
  • $\sin$ is entire
  • $\sin(z)={\frac {\exp({\acute {i}}z)-\exp(-{\acute {i}}z)}{2{\acute {i}}}}$
  • Complex reciprocal of sine ($\csc(z)$)
  • Complex sine of reciprocal ($\sin(z^{-1})$) (complex map / 3D plot)
  • Complex natural logarithm of sine ($\log(\sin(z))$) (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • Complex sine of natural logarithm ($\sin(\log(z))$) (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • Complex inverse of sine ($\arcsin(z)$)
  • Complex derivative of sine ($\cos(z)$)
  • Complex integral of sine ($-\cos(z)+C$) (complex map / 3D plot)
  • Complex sine of sine ($\sin(\sin(z))$) (complex map / 3D plot)
• Dual sine (dual map / 3D plot)
• Split-complex sine (split-complex map / 3D plot)
• Quaternionic sine (vectors / sliders / 3D rotations)
• Octonionic sine (vectors / sliders)

Cosine

Complex $\cos(z)$ contour plot

Complex $\cos(z)$ domain colouring

Dual $\cos(z)$ contour plot

Dual $\cos(z)$ domain colouring

Split-complex $\cos(z)$ contour plot

Split-complex $\cos(z)$ domain colouring

Complex $\cos(z^{-1})$ contour plot

Complex $\cos(z^{-1})$ domain colouring

Complex $\log(\cos(z))$ contour plot

Complex $\log(\cos(z))$ domain colouring

Complex $\cos(\log(z))$ contour plot. Branch cut is $(-\infty ,0]$.

Complex $\cos(\log(z))$ domain colouring. Branch cut is $(-\infty ,0]$.

Complex $\cos(\cos(z))$ contour plot

Complex $\cos(\cos(z))$ domain colouring

$\cos(z):=\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n}}{(2n)!}}$

• Complex cosine (complex map / 3D plot / Mandelbrot / iterates)
  • $\cos :{\mathbb {C} }\to {\mathbb {C} }$
  • $\cos$ is elementary
  • $\cos$ is complex analytic
  • $\cos$ has simple zeros at all $n\pi +{\frac {\pi }{2}}:n\in {\mathbb {Z} }$
  • $\cos$ has an essential singularity at $\infty$
  • $\cos$ is conformal except at elements of $\{\pi n|n\in {\mathbb {Z} }\}$
  • $\cos$ is entire
  • $\cos(z)={\frac {\exp({\acute {i}}z)+\exp(-{\acute {i}}z)}{2}}$
  • Complex reciprocal of cosine ($\sec(z)$)
  • Complex cosine of reciprocal ($\cos(z^{-1})$) (complex map / 3D plot)
  • Complex natural logarithm of cosine ($\log(\cos(z))$) (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • Complex cosine of natural logarithm ($\cos(\log(z))$) (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • Complex inverse of cosine ($\arccos(z)$)
  • Complex derivative of cosine ($-\sin(z)$) (complex map / 3D plot)
  • Complex integral of cosine ($\sin(z)+C$) (complex map / 3D plot)
  • Complex cosine of cosine ($\cos(\cos(z))$) (complex map / 3D plot)
• Dual cosine (dual map / 3D plot)
• Split-complex cosine (split-complex map / 3D plot)
• Quaternionic cosine (vectors / sliders / 3D rotations)
• Octonionic cosine (vectors / sliders)

Tangent

Complex $\tan(z)$ contour plot

Complex $\tan(z)$ domain colouring

Dual $\tan(z)$ contour plot

Dual $\tan(z)$ domain colouring

Split-complex $\tan(z)$ contour plot

Split-complex $\tan(z)$ domain colouring

Complex $\tan(z^{-1})$ contour plot

Complex $\tan(z^{-1})$ domain colouring

Complex $\log(\tan(z))$ contour plot

Complex $\log(\tan(z))$ domain colouring

Complex $\tan(\log(z))$ contour plot

Complex $\tan(\log(z))$ domain colouring

Complex $\tan(\tan(z))$ contour plot

Complex $\tan(\tan(z))$ domain colouring

Dual tan(tan(z)).png

Dual $\tan(\tan(z))$ contour plot

Dual tan(tan(z)) dc.png

Dual $\tan(\tan(z))$ domain colouring

Split-complex $\tan(\tan(z))$ contour plot

Split-complex $\tan(\tan(z))$ domain colouring

$\tan(z):=\sin(z)\cdot \sec(z)$

• Complex tangent (complex map / 3D plot / iterates)
  • $\tan :{\mathbb {C} }\backslash \{n\pi +{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}\to {\mathbb {C} }\backslash \{-{\acute {i}},{\acute {i}}\}$
  • $\tan$ is elementary
  • $\tan$ is analytic on its domain
  • $\tan$ is meromorphic
  • $\tan$ has simple zeros at all $n\pi :n\in {\mathbb {Z} }$
  • $\tan$ has simple poles at all $n\pi +{\frac {\pi }{2}}:n\in {\mathbb {Z} }$
  • $\tan$ has an essential singularity at $\infty$
  • $\tan$ is conformal
  • Complex reciprocal of tangent ($\cot(z)$)
  • Complex tangent of reciprocal ($\tan(z^{-1})$) (complex map / 3D plot)
  • Complex inverse of tangent ($\arctan(z)$)
  • Complex derivative of tangent ($\sec ^{2}(z)$) (complex map / 3D plot)
  • Complex integral of tangent ($-\log(\cos(z))+C$) (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • Complex tangent of tangent ($\tan(\tan(z))$) (complex map / 3D plot)
• Dual tangent (dual map / 3D plot)
• Split-complex tangent (split-complex map / 3D plot)
• Quaternionic tangent (vectors / sliders / 3D rotations)
• Octonionic tangent (vectors / sliders)

Cosecant

Complex $\csc(z)$ contour plot

Complex $\csc(z)$ domain colouring

Dual csc(z).png

Dual $\csc(z)$ contour plot

Dual csc(z) dc.png

Dual $\csc(z)$ domain colouring

Split-complex csc(z).png

Split-complex $\csc(z)$ contour plot

Split-complex csc(z) dc.png

Split-complex $\csc(z)$ domain colouring

$\csc(z):=(\sin(z))^{-1}$

• Complex cosecant (complex map / 3D plot)
  • $\csc :{\mathbb {C} }\backslash \{\pi n|n\in {\mathbb {Z} }\}\to {\mathbb {C} }\backslash \{0\}$
  • $\csc$ is elementary
  • $\csc$ is analytic on any open set on its domain
  • $\csc$ has simple poles at all $n\pi :n\in {\mathbb {Z} }$
  • $\csc$ has an essential singularity at $\infty$
  • $\csc$ is conformal
• Dual cosecant (dual map / 3D plot)
• Split-complex cosecant (split-complex map / 3D plot)
• Quaternionic cosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic cosecant ([ vectors] / [ sliders])

Secant

Complex $\sec(z)$ contour plot

Complex $\sec(z)$ domain colouring

Dual $\sec(z)$ contour plot

Dual $\sec(z)$ domain colouring

Split-complex $\sec(z)$ contour plot

Split-complex $\sec(z)$ domain colouring

Complex $\sec(z^{-1})$ contour plot

Complex $\sec(z^{-1})$ domain colouring

Complex $\log(\sec(z))$ contour plot

Complex $\log(\sec(z))$ domain colouring

Complex $\sec(\log(z))$ contour plot. Branch cut is $(-\infty ,0]$.

Complex $\sec(\log(z))$ domain colouring. Branch cut is $(-\infty ,0]$.

Complex $\sec(\sec(z))$ contour plot

Complex $\sec(\sec(z))$ domain colouring

$\sec(z):=(\cos(z))^{-1}$

• Complex secant (complex map / 3D plot / iterates)
  • $\sec :{\mathbb {C} }\backslash \{\pi n+{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}\to {\mathbb {C} }\backslash \{0\}$
  • $\sec$ is elementary
  • $\sec$ is analytic on any open set on its domain
  • $\sec$ has simple poles at all $n\pi +{\frac {\pi }{2}}:n\in {\mathbb {Z} }$
  • $\sec$ has an essential singularity at $\infty$
  • $\sec$ is conformal
• Dual secant (dual map / 3D plot)
• Split-complex secant (split-complex map / 3D plot)
• Quaternionic secant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic secant ([ vectors] / [ sliders])

Cotangent

Complex $\cot(z)$ contour plot

Complex $\cot(z)$ domain colouring

Dual cot(z).png

Dual $\cot(z)$ contour plot

Dual cot(z) dc.png

Dual $\cot(z)$ domain colouring

Split-complex cot(z).png

Split-complex $\cot(z)$ contour plot

Split-complex cot(z) dc.png

Split-complex $\cot(z)$ domain colouring

$\cot(z):=(\tan(z))^{-1}$

• Complex cotangent (complex map / 3D plot)
  • $\cot :{\mathbb {C} }\backslash \{\pi n|n\in {\mathbb {Z} }\}\to {\mathbb {C} }$
  • $\cot$ is elementary
  • $\cot$ is analytic on any open set on its domain
  • $\cot$ has simple poles at all $n\pi :n\in {\mathbb {Z} }$
  • $\cot$ has an essential singularity at $\infty$
  • $\cot$ is conformal
• Dual cotangent (dual map / 3D plot)
• Split-complex cotangent (split-complex map / 3D plot)
• Quaternionic cotangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic cotangent ([ vectors] / [ sliders])

Cis

Complex ${\rm {cis}}(z)$ contour plot

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

• Complex cis (complex map / 3D plot)
  • ${\rm {cis}}:{\mathbb {C} }\to {\mathbb {C} }\backslash \{0\}$
  • ${\rm {cis}}$ is elementary
  • ${\rm {cis}}$ is complex analytic
  • ${\rm {cis}}$ is conformal
  • ${\rm {cis}}$ is entire
  • ${\rm {cis}}(z)=\cos(z)+{\acute {i}}\sin(z)$
  • ${\rm {cis}}(z)=\exp({\acute {i}}z)$

Cocis

Complex ${\rm {cocis}}(z)$ contour plot

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

• Complex cocis (complex map / 3D plot)
  • ${\rm {cocis}}:{\mathbb {C} }\to {\mathbb {C} }\backslash \{0\}$
  • ${\rm {cocis}}$ is elementary
  • ${\rm {cocis}}$ is complex analytic
  • ${\rm {cocis}}$ is conformal
  • ${\rm {cocis}}$ is entire
  • ${\rm {cocis}}(z)=\sin(z)+{\acute {i}}\cos(z)$
  • ${\rm {cocis}}(z)={\acute {i}}\exp(-{\acute {i}}z)$

Inverse sine

Complex $\arcsin(z)$ contour plot

Complex $\arcsin(z)$ domain colouring

Littrow projection of the Earth (can be defined from a complex map of $\sin$/contour plot of $\arcsin$).

• Complex inverse sine (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
  • $\arcsin :{\mathbb {C} }\to {\mathbb {C} }$
  • $\arcsin$ is elementary
  • $\arcsin$ is analytic except at $-1$ and $1$
  • $\arcsin$ has a simple zero at $0$
  • $\arcsin$ is conformal except at $-1$ and $1$
  • $\arcsin$ is multivalued
  • $\arcsin$ has branch points at $-1$ and $1$
  • $\arcsin(z)=-{\acute {i}}\log({\sqrt {1-z^{2}}}+{\acute {i}}z)$
• Dual inverse sine ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse sine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse sine (vectors of principal value / sliders of principal value / 3D rotations of principal value)
  • $\arcsin$ is extremely multivalued for elements of $\{x\in {\mathbb {R} }:1<|x|\}$
• Octonionic inverse sine (vectors of principal value / sliders of principal value)
  • $\arcsin$ is extremely multivalued for elements of $\{x\in {\mathbb {R} }:1<|x|\}$

Inverse cosine

Complex $\arccos(z)$ contour plot

Complex $\arccos(z)$ domain colouring

• Complex inverse cosine (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
  • $\arccos :{\mathbb {C} }\to {\mathbb {C} }$
  • $\arccos$ is elementary
  • $\arccos$ is analytic except at $-1$ and $1$
  • $\arccos$ has a zero of order ${\frac {1}{2}}$ at $1$
  • $\arccos$ is conformal except at $-1$ and $1$
  • $\arccos$ is multivalued
  • $\arccos$ has branch points at $-1$ and $1$
  • $\arccos(z)={\frac {\pi }{2}}+{\acute {i}}\log({\sqrt {1-z^{2}}}+{\acute {i}}z)$
• Dual inverse cosine ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse cosine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse cosine (vectors of principal value / sliders of principal value / 3D rotations of principal value)
  • $\arccos$ is extremely multivalued for elements of $\{x\in {\mathbb {R} }:1<|x|\}$
• Octonionic inverse cosine (vectors of principal value / sliders of principal value)
  • $\arccos$ is extremely multivalued for elements of $\{x\in {\mathbb {R} }:1<|x|\}$

Inverse tangent

Complex $\arctan(z)$ contour plot. Branch cuts are $[1,\infty ){\acute {i}}$ and $(-\infty ,-1]{\acute {i}}$.

Complex $\arctan(z)$ domain colouring. Branch cuts are $[1,\infty ){\acute {i}}$ and $(-\infty ,-1]{\acute {i}}$.

• Complex inverse tangent (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
  • $\arctan :{\mathbb {C} }\backslash \{-{\acute {i}},{\acute {i}}\}\to {\mathbb {C} }\backslash \{n\pi +{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}$
  • $\arctan$ is elementary
  • $\arctan$ is analytic on any open set on its domain
  • $\arctan$ has a simple zero at $0$
  • $\arctan$ is conformal
  • $\arctan$ is multivalued
  • $\arctan$ has branch points at $-{\acute {i}}$ and ${\acute {i}}$
  • $\arctan(z)={\frac {{\acute {i}}(\log(1-{\acute {i}}z-\log(1+{\acute {i}}z))}{2}}$
• Complex two-argument inverse tangent (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• Dual inverse tangent ([ dual map of principal branch] / [ 3D plot of principal branch])
• Dual two-argument inverse tangent ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse tangent ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Split-complex two-argument inverse tangent ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse tangent (vectors of principal value / sliders of principal value / 3D rotations of principal value)
• Quaternionic two-argument inverse tangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse tangent (vectors of principal value / sliders of principal value)
• Octonionic two-argument inverse tangent ([ vectors] / [ sliders])

Inverse cosecant

Complex ${\rm {arccsc}}(z)$ contour plot

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

• Complex inverse cosecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arccsc}}:{\mathbb {C} }\backslash \{0\}\to {\mathbb {C} }\backslash \{0\}$
  • ${\rm {arccsc}}$ is elementary
  • ${\rm {arccsc}}$ is analytic everywhere in its domain except at $-1$ and $1$
  • ${\rm {arccsc}}$ is conformal except at $-1$ and $1$
  • ${\rm {arccsc}}$ is multivalued
  • ${\rm {arccsc}}$ has branch points at $-1$ and $0$
  • ${\rm {arccsc}}(z)=-{\acute {i}}\log({\sqrt {1-z^{-2}}}+{\acute {i}}z^{-1})$
• Dual inverse cosecant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse cosecant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse cosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse cosecant ([ vectors] / [ sliders])

Inverse secant

Complex ${\rm {arcsec}}(z)$ contour plot

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

• Complex inverse secant (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
  • ${\rm {arcsec}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {arcsec}}$ is elementary
  • ${\rm {arcsec}}$ is analytic except at $-1$ and $1$
  • ${\rm {arcsec}}$ is conformal except at $-1$ and $1$
  • ${\rm {arcsec}}$ is multivalued
  • ${\rm {arcsec}}(z)={\frac {\pi }{2}}+{\acute {i}}\log({\sqrt {1-z^{-2}}}+{\acute {i}}z^{-1})$
• Dual inverse secant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse secant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse secant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse secant ([ vectors] / [ sliders])

Inverse cotangent

Complex ${\rm {arccot}}(z)$ contour plot.

Complex ${\rm {arccot}}(z)$ domain colouring.

Complex ${\rm {arccot}}(z)$ (alternate principal branch) contour plot.

Complex ${\rm {arccot}}(z)$ (alternate principal branch) contour plot.

${\rm {arccot}}(z):={\frac {\pi }{2}}-\arctan(z)$

• Complex inverse cotangent (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arccot}}:{\mathbb {C} }\backslash \{-{\acute {i}},{\acute {i}}\}\to {\mathbb {C} }\backslash \{n\pi +{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}$
  • ${\rm {arccot}}$ is elementary
  • ${\rm {arccot}}$ is analytic on any open set on its domain
  • ${\rm {arccot}}$ is conformal
  • ${\rm {arccot}}$ is multivalued
• Dual inverse cotangent ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse cotangent ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse cotangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse cotangent ([ vectors] / [ sliders])

Inverse cis

Complex arccis(z).png

Complex ${\rm {arccis}}(z)$ contour plot

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

• Complex inverse cis (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arccis}}:{\mathbb {C} }\backslash \{0\}\to {\mathbb {C} }$
  • ${\rm {arccis}}$ is elementary
  • ${\rm {arccis}}$ is complex analytic
  • ${\rm {arccis}}$ is conformal
  • ${\rm {arccis}}$ is multivalued
  • ${\rm {arccis}}(z)=-{\acute {i}}\log(z)$

Inverse cocis

Complex arccocis(z).png

Complex ${\rm {arccocis}}(z)$ contour plot

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

• Complex inverse cocis (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arccocis}}:{\mathbb {C} }\backslash \{0\}\to {\mathbb {C} }$
  • ${\rm {arccocis}}$ is elementary
  • ${\rm {arccocis}}$ is complex analytic
  • ${\rm {arccocis}}$ is conformal
  • ${\rm {arccocis}}$ is multivalued
  • ${\rm {arccocis}}(z)={\acute {i}}\log(-{\acute {i}}z)$

Versine

Complex ${\rm {vers}}(z)$ contour plot

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

${\rm {vers}}(z):=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}z^{2n}}{(2n)!}}=1-\cos(z)$

• Complex versine (complex map / 3D plot)
  • ${\rm {vers}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {vers}}$ is elementary
  • ${\rm {vers}}$ is complex analytic
  • ${\rm {vers}}$ is conformal except at elements of $\{\pi n|n\in {\mathbb {Z} }\}$
  • ${\rm {vers}}$ is entire
• Dual versine (dual map / 3D plot)
• Split-complex versine (split-complex map / 3D plot)
• Quaternionic versine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic versine ([ vectors] / [ sliders])

Coversine

Complex covers(z).png

Complex ${\rm {covers}}(z)$ contour plot

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

• Complex coversine (complex map / 3D plot)
  • ${\rm {covers}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {covers}}$ is elementary
  • ${\rm {covers}}$ is complex analytic
  • ${\rm {covers}}$ is conformal except at elements of $\{\pi n+{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}$
  • ${\rm {covers}}$ is entire
• Dual coversine (dual map / 3D plot)
• Split-complex coversine (split-complex map / 3D plot)
• Quaternionic coversine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic coversine ([ vectors] / [ sliders])

Haversine

Complex ${\rm {havers}}(z)$ contour plot

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

${\rm {havers}}(z):=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}z^{2n}}{(2n)!\cdot 2}}={\frac {{\rm {vers}}(z)}{2}}$

• Complex haversine (complex map / 3D plot)
  • ${\rm {havers}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {havers}}$ is elementary
  • ${\rm {havers}}$ is complex analytic
  • ${\rm {havers}}$ is conformal except at elements of $\{\pi n|n\in {\mathbb {Z} }\}$
  • ${\rm {havers}}$ is entire
• Dual haversine (dual map / 3D plot)
• Split-complex haversine (split-complex map / 3D plot)
• Quaternionic haversine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic haversine ([ vectors] / [ sliders])

Excosecant

Complex excsc(z).png

Complex ${\rm {excsc}}(z)$ contour plot

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

• Complex excosecant (complex map / 3D plot)
  • ${\rm {excsc}}:{\mathbb {C} }\backslash \{\pi n|n\in {\mathbb {Z} }\}\to {\mathbb {C} }\backslash \{-1\}$
  • ${\rm {excsc}}$ is elementary
  • ${\rm {excsc}}$ is analytic on any open set on its domain
  • ${\rm {excsc}}$ is conformal
• Dual excosecant (dual map / 3D plot)
• Split-complex excosecant (split-complex map / 3D plot)
• Quaternionic excosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic excosecant ([ vectors] / [ sliders])

Exsecant

Complex exsec(z).png

Complex ${\rm {exsec}}(z)$ contour plot

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

• Complex exsecant (complex map / 3D plot)
  • ${\rm {exsec}}:{\mathbb {C} }\backslash \{\pi n+{\frac {\pi }{2}}|n\in {\mathbb {Z} }\}\to {\mathbb {C} }\backslash \{-1\}$
  • ${\rm {exsec}}$ is elementary
  • ${\rm {exsec}}$ is analytic on any open set on its domain
  • ${\rm {exsec}}$ is conformal
• Dual exsecant (dual map / 3D plot)
• Split-complex exsecant (split-complex map / 3D plot)
• Quaternionic exsecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic exsecant ([ vectors] / [ sliders])

Chord

Complex crd(z).png

Complex ${\rm {crd}}(z)$ contour plot

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

• Complex chord ([ complex map] / [ 3D plot])
  • ${\rm {crd}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {crd}}$ is elementary
  • ${\rm {crd}}$ is complex analytic
  • ${\rm {crd}}$ is conformal
• Dual chord ([ dual map] / [ 3D plot])
• Split-complex chord ([ split-complex map] / [ 3D plot])
• Quaternionic chord ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic chord ([ vectors] / [ sliders])

Inverse versine

Complex arcvers(z).png

Complex ${\rm {arcvers}}(z)$ contour plot

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

• Complex inverse versine (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arcvers}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {arcvers}}$ is elementary
  • ${\rm {arcvers}}$ is analytic except at $0$ and $2$
  • ${\rm {arcvers}}$ is conformal except at $0$ and $2$
  • ${\rm {arcvers}}$ is multivalued
• Dual inverse versine ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse versine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse versine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse versine ([ vectors] / [ sliders])

Inverse coversine

Complex arccovers(z).png

Complex ${\rm {arccovers}}(z)$ contour plot

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

• Complex inverse coversine (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arccovers}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {arccovers}}$ is elementary
  • ${\rm {arccovers}}$ is analytic except at $0$ and $2$
  • ${\rm {arccovers}}$ is conformal except at $0$ and $2$
  • ${\rm {arccovers}}$ is multivalued
• Dual inverse versine ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse coversine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse coversine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse coversine ([ vectors] / [ sliders])

Inverse haversine

Complex archavers(z).png

Complex ${\rm {archavers}}(z)$ contour plot

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

• Complex inverse haversine (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {archavers}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {archavers}}$ is elementary
  • ${\rm {archavers}}$ is analytic except at $0$ and $2$
  • ${\rm {archavers}}$ is conformal except at $0$ and $1$
  • ${\rm {archavers}}$ is multivalued
• Dual inverse haversine ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse haversine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse haversine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse haversine ([ vectors] / [ sliders])

Inverse excosecant

Complex arcexcsc(z).png

Complex ${\rm {arcexcsc}}(z)$ contour plot

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

• Complex inverse excosecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arcexcsc}}:{\mathbb {C} }\backslash \{-1\}\to {\mathbb {C} }\backslash \{0\}$
  • ${\rm {arcexcsc}}$ is elementary
  • ${\rm {arcexcsc}}$ is analytic everywhere in its domain except at $-2$ and $0$
  • ${\rm {arcexcsc}}$ is conformal except at $-2$ and $0$
  • ${\rm {arcexcsc}}$ is multivalued
  • ${\rm {arcexcsc}}$ has branch points at $-1$ and $0$
• Dual inverse excosecant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse excosecant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse excosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse excosecant ([ vectors] / [ sliders])

Inverse exsecant

Complex arcexsec(z).png

Complex ${\rm {arcexsec}}(z)$ contour plot

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

• Complex inverse exsecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
  • ${\rm {arcexcsc}}:{\mathbb {C} }\backslash \{-1\}\to {\mathbb {C} }\backslash \{0\}$
  • ${\rm {arcexsec}}$ is elementary
  • ${\rm {arcexsec}}$ is analytic everywhere in its domain except at $-2$ and $0$
  • ${\rm {arcexsec}}$ is conformal except at $-2$ and $0$
  • ${\rm {arcexsec}}$ is multivalued
  • ${\rm {arcexsec}}$ has branch points at $-1$ and $0$
• Dual inverse exsecant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse exsecant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse exsecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse exsecant ([ vectors] / [ sliders])

Inverse chord

Complex arccrd(z).png

Complex ${\rm {arccrd}}(z)$ contour plot

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

• Complex inverse chord ([ complex map of principal branch] / [ 3D plot of principal branch] / [ Riemann surface])
  • ${\rm {arccrd}}:{\mathbb {C} }\to {\mathbb {C} }$
  • ${\rm {arccrd}}$ is elementary
  • ${\rm {arccrd}}$ is analytic everywhere in its domain except at $-2$ and $2$
  • ${\rm {arccrd}}$ is conformal except at $-2$ and $2$
  • ${\rm {arccrd}}$ is multivalued
  • ${\rm {arccrd}}$ has branch points at $-2$ and $2$
• Dual inverse chord ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse chord ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse chord ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse chord ([ vectors] / [ sliders])

Schläflian function

Complex ${\rm {Sch}}(\{z\})$ contour plot.

Complex schläflian(7/3,z).png

Complex ${\rm {Sch}}(\{{\frac {7}{3}},z\})$ contour plot.

Complex schläflian(5/2,z).png

Complex ${\rm {Sch}}(\{{\frac {5}{2}},z\})$ contour plot.

Complex ${\rm {Sch}}(\{3,z\})$ contour plot.

Complex schläflian(7/2,z).png

Complex ${\rm {Sch}}(\{{\frac {7}{2}},z\})$ contour plot.

Complex ${\rm {Sch}}(\{4,z\})$ contour plot.

Complex ${\rm {Sch}}(\{5,z\})$ contour plot.

Complex schläflian(6,z).png

Complex ${\rm {Sch}}(\{6,z\})$ contour plot.

Complex schläflian(7,z).png

Complex ${\rm {Sch}}(\{7,z\})$ contour plot.

Complex schläflian(8,z).png

Complex ${\rm {Sch}}(\{8,z\})$ contour plot.

Complex ${\rm {Sch}}(\{\infty ,z\})$ contour plot.

Complex ${\rm {Sch}}(\{12{\acute {i}},z\})$ contour plot.

Complex ${\rm {Sch}}(\{9{\acute {i}},z\})$ contour plot.

Complex schläflian(6i,z).png

Complex ${\rm {Sch}}(\{6{\acute {i}},z\})$ contour plot.

Complex schläflian(3i,z).png

Complex ${\rm {Sch}}(\{3{\acute {i}},z\})$ contour plot.

Complex schläflian(i,z).png

Complex ${\rm {Sch}}(\{{\acute {i}},z\})$ contour plot.

Complex schläflian(5/2,3,z).png

Complex ${\rm {Sch}}(\{{\frac {5}{2}},3,z\})$ contour plot.

Complex schläflian(5/2,5,z).png

Complex ${\rm {Sch}}(\{{\frac {5}{2}},5,z\})$ contour plot.

Complex schläflian(3,3,z).png

Complex ${\rm {Sch}}(\{3,{\frac {5}{2}},z\})$ contour plot.

Complex schläflian(3,3,z).png

Complex ${\rm {Sch}}(\{3,3,z\})$ contour plot.

Complex schläflian(3,4,z).png

Complex ${\rm {Sch}}(\{3,4,z\})$ contour plot.

Complex schläflian(4,3,z).png

Complex ${\rm {Sch}}(\{4,3,z\})$ contour plot.

Complex schläflian(5,3,z).png

Complex ${\rm {Sch}}(\{5,3,z\})$ contour plot.

Complex schläflian(3,6,z).png

Complex ${\rm {Sch}}(\{3,6,z\})$ contour plot.

Complex schläflian(4,4,z).png

Complex ${\rm {Sch}}(\{4,4,z\})$ contour plot.

Complex schläflian(6,3,z).png

Complex ${\rm {Sch}}(\{6,3,z\})$ contour plot.

Complex schläflian(3,7,z).png

Complex ${\rm {Sch}}(\{3,7,z\})$ contour plot.

Complex schläflian(4,5,z).png

Complex ${\rm {Sch}}(\{4,5,z\})$ contour plot.

Complex schläflian(5,4,z).png

Complex ${\rm {Sch}}(\{5,4,z\})$ contour plot.

Complex schläflian(6,4,z).png

Complex ${\rm {Sch}}(\{6,4,z\})$ contour plot.

Complex schläflian(7,3,z).png

Complex ${\rm {Sch}}(\{7,3,z\})$ contour plot.

Complex schläflian(8,3,z).png

Complex ${\rm {Sch}}(\{8,3,z\})$ contour plot.

${\rm {Sch}}(\{z\})=4\sin ^{2}({\frac {\pi }{z}})$

• Complex polygonal Schläflian function (complex map / 3D plot)
  • ${\rm {Sch}}:{\mathbb {C} }\backslash \{0\}\to {\mathbb {C} }$
  • ${\rm {Sch}}$ is elementary
  • ${\rm {Sch}}$ is analytic everywhere in its domain
  • ${\rm {Sch}}$ is conformal except at elements of $\{{\frac {2}{n}}|n\in {\mathbb {Z} }\}$
  • ${\rm {Sch}}$ is multivalued
• Dual polygonal Schläflian function ([ dual map] / [ 3D plot])
• Split-complex polygonal Schläflian function ([ split-complex map] / [ 3D plot])
• Quaternionic polygonal Schläflian function ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic polygonal Schläflian function ([ vectors] / [ sliders])

References

Graphs

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