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
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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
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Inverse 1/x,
rational function (x+4)/(x5-3ix3+2) | Complex inverse 1/x | Complex rational function (x+4)/(x5-3ix3+2) |
Nonanalytic functions
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Absolute value |x|,
floor function ⌊x⌋ | Complex absolute value |x| | Complex floor function ⌊x⌋ |
Roots
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Square root x1/2,
cube root x1/3 | Complex square root x1/2 | Complex cube root x1/3 |
Exponentiation
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Exponential function exp(x),
natural logarithm ln(x) | Complex exponential function exp(x) | Complex natural logarithm ln(x) |
Hyper-exponentiation
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2-tetration xx,
3-tetration x(xx) | Complex 2-tetration xx | Complex 3-tetration x(xx) |
Lambert W function
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W0(x) (branch 0),
W−1(x) (branch −1) | Complex W0(x) (branch 0) | Complex W−1(x) (branch −1) |
Trigonometric functions 1
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Sine,
cosine | Complex sine | Complex cosine |
Trigonometric functions 2
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Tangent,
cotangent | Complex tangent | Complex cotangent |
Trigonometric functions 3
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Secant,
cosecant | Complex secant | Complex cosecant |
Inverse trigonometric functions 1
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Inverse sine,
inverse cosine | Complex inverse sine | Complex inverse cosine |
Inverse trigonometric functions 2
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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
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Gaussian, exp(−x2),
error function erf(x) | Complex Gaussian, exp(−x2) | Complex error function erf(x) |
Fresnel integrals
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Fresnel integral S(x),
Fresnel integral C(x) | Complex Fresnel integral S(x) | Complex Fresnel integral C(x) |
Exponential integrals
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Exponential integral Ei(x),
Logarithmic integral li(x) | Complex exponential integral Ei(x) | Complex Logarithmic integral li(x) |
Trigonometric integrals
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Sine integral Si(x),
cosine integral Ci(x) | Complex sine integral Si(x) | Complex cosine integral Ci(x) |
Hyperbolic integrals
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Hyperbolic sine integral Shi(x),
hyperbolic cosine integral Chi(x) | Complex hyperbolic sine integral Shi(x) | Complex hyperbolic cosine integral Chi(x) |
Airy functions
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Airy function Ai(x),
Airy function Bi(x) | Complex Airy function Ai(x) | Complex Airy function Bi(x) |
Bessel function of the first kind
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Bessel function J0(x),
Bessel function J1(x) | Complex Bessel function J0(x) | Complex Bessel function J1(x) |
Gamma function
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| Gamma function Γ(x) | Complex gamma function Γ(x) |
Polygamma functions
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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
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| Riemann zeta function ζ(x) | Complex Riemann zeta function ζ(x) |
Jacobi theta functions 1-2
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θ1(x,1/2),
θ2(x,1/2) | Complex θ1(x,1/2) | Complex θ2(x,1/2) |
Jacobi theta functions 3-4
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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
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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
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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
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| Complex θ3((1+i)/3,q) | Complex θ4((1+i)/3,q) |
Derivative of Jacobi theta functions 1-2, variable nome
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| Complex θ1((1+i)/3,q) | Complex θ2((1+i)/3,q) |
Derivative of Jacobi theta functions 3-4, variable nome
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| Complex θ'3((1+i)/3,q) | Complex θ'4((1+i)/3,q) |