Gallery of mathematical functions

Generated with mpmath and matplotlib by this script.

The complex plots show the magnitude as brightness (0 = black, ∞ = white) and the phase as hue (positive real = red, imaginary = lime, negative real = cyan, negative imaginary = violet).


Linear function f(x) = x,
cubic, 0.1 x3 - x + 1
Complex linear function f(x) = xComplex cubic, 0.1 x3 - x + 1

Rational functions

Inverse 1/x,
rational function (x+4)/(x5-3ix3+2)
Complex inverse 1/xComplex rational function (x+4)/(x5-3ix3+2)

Nonanalytic functions

Absolute value |x|,
floor function ⌊x⌋
Complex absolute value |x|Complex floor function ⌊x⌋


Square root x1/2,
cube root x1/3
Complex square root x1/2Complex cube root x1/3


Exponential function exp(x),
natural logarithm ln(x)
Complex exponential function exp(x)Complex natural logarithm ln(x)


2-tetration xx,
3-tetration x(xx)
Complex 2-tetration xxComplex 3-tetration x(xx)

Lambert W function

W0(x) (branch 0),
W−1(x) (branch −1)
Complex W0(x) (branch 0)Complex W−1(x) (branch −1)

Trigonometric functions 1

Complex sineComplex cosine

Trigonometric functions 2

Complex tangentComplex cotangent

Trigonometric functions 3

Complex secantComplex cosecant

Inverse trigonometric functions 1

Inverse sine,
inverse cosine
Complex inverse sineComplex inverse cosine

Inverse trigonometric functions 2

Inverse tangent,
inverse cotangent
Complex inverse tangentComplex inverse cotangent

Inverse trigonometric functions 3

Inverse secant,
inverse cosecant
Complex inverse secantComplex inverse cosecant

Hyperbolic functions 1

Hyperbolic sine,
hyperbolic cosine
Complex hyperbolic sineComplex hyperbolic cosine

Hyperbolic functions 2

Hyperbolic tangent,
hyperbolic cotangent
Complex hyperbolic tangentComplex hyperbolic cotangent

Hyperbolic functions 3

Hyperbolic secant,
hyperbolic cosecant
Complex hyperbolic secantComplex hyperbolic cosecant

Inverse hyperbolic functions 1

Inverse hyperbolic sine,
inverse hyperbolic cosine
Complex inverse hyperbolic sineComplex inverse hyperbolic cosine

Inverse hyperbolic functions 2

Inverse hyperbolic tangent,
inverse hyperbolic cotangent
Complex inverse hyperbolic tangentComplex inverse hyperbolic cotangent

Inverse hyperbolic functions 3

Inverse hyperbolic secant,
inverse hyperbolic cosecant
Complex inverse hyperbolic secantComplex inverse hyperbolic cosecant

Error function

Gaussian, exp(−x2),
error function erf(x)
Complex Gaussian, exp(−x2)Complex error function erf(x)

Fresnel integrals

Fresnel integral S(x),
Fresnel integral C(x)
Complex Fresnel integral S(x)Complex Fresnel integral C(x)

Exponential integrals

Exponential integral Ei(x),
Logarithmic integral li(x)
Complex exponential integral Ei(x)Complex Logarithmic integral li(x)

Trigonometric integrals

Sine integral Si(x),
cosine integral Ci(x)
Complex sine integral Si(x)Complex cosine integral Ci(x)

Hyperbolic integrals

Hyperbolic sine integral Shi(x),
hyperbolic cosine integral Chi(x)
Complex hyperbolic sine integral Shi(x)Complex hyperbolic cosine integral Chi(x)

Airy functions

Airy function Ai(x),
Airy function Bi(x)
Complex Airy function Ai(x)Complex Airy function Bi(x)

Bessel function of the first kind

Bessel function J0(x),
Bessel function J1(x)
Complex Bessel function J0(x)Complex Bessel function J1(x)

Gamma function

Gamma function Γ(x)Complex gamma function Γ(x)

Polygamma functions

Digamma function ψ(0)(x),
trigamma function ψ(1)(x)
Complex digamma function ψ(0)(x)Complex trigamma function ψ(1)(x)

Riemann zeta function

Riemann zeta function ζ(x)Complex Riemann zeta function ζ(x)

Jacobi theta functions 1-2

Complex θ1(x,1/2)Complex θ2(x,1/2)

Jacobi theta functions 3-4

Complex θ3(x,1/2)Complex θ4(x,1/2)

Derivative of Jacobi theta functions 1-2

Complex θ'1(x,1/2)Complex θ'2(x,1/2)

Derivative of Jacobi theta functions 3-4

Complex θ'3(x,1/2)Complex θ'4(x,1/2)

Jacobi theta functions 1-2, variable nome

Complex θ1((1+i)/3,q)Complex θ2((1+i)/3,q)

Jacobi theta functions 3-4, variable nome

Complex θ3((1+i)/3,q)Complex θ4((1+i)/3,q)

Derivative of Jacobi theta functions 1-2, variable nome

Complex θ1((1+i)/3,q)Complex θ2((1+i)/3,q)

Derivative of Jacobi theta functions 3-4, variable nome

Complex θ'3((1+i)/3,q)Complex θ'4((1+i)/3,q)