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).

Polynomials

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

Rational functions

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

Nonanalytic functions

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

Roots

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

Exponentiation

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

Hyper-exponentiation

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

 Sine,cosine Complex sine Complex cosine

Trigonometric functions 2

 Tangent,cotangent Complex tangent Complex cotangent

Trigonometric functions 3

 Secant,cosecant Complex secant Complex cosecant

Inverse trigonometric functions 1

 Inverse sine,inverse cosine Complex inverse sine Complex inverse cosine

Inverse trigonometric functions 2

 Inverse tangent,inverse cotangent Complex inverse tangent Complex inverse cotangent

Inverse trigonometric functions 3

 Inverse secant,inverse cosecant Complex inverse secant Complex inverse cosecant

Hyperbolic functions 1

 Hyperbolic sine,hyperbolic cosine Complex hyperbolic sine Complex hyperbolic cosine

Hyperbolic functions 2

 Hyperbolic tangent,hyperbolic cotangent Complex hyperbolic tangent Complex hyperbolic cotangent

Hyperbolic functions 3

 Hyperbolic secant,hyperbolic cosecant Complex hyperbolic secant Complex hyperbolic cosecant

Inverse hyperbolic functions 1

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

Inverse hyperbolic functions 2

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

Inverse hyperbolic functions 3

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

 θ1(x,1/2),θ2(x,1/2) Complex θ1(x,1/2) Complex θ2(x,1/2)

Jacobi theta functions 3-4

 θ3(x,1/2),θ4(x,1/2) Complex θ3(x,1/2) Complex θ4(x,1/2)

Derivative of Jacobi theta functions 1-2

 θ'1(x,1/2),θ'2(x,1/2) Complex θ'1(x,1/2) Complex θ'2(x,1/2)

Derivative of Jacobi theta functions 3-4

 θ'3(x,1/2),θ'4(x,1/2) 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)