Examples for

Special Functions

Special functions refer to mathematical functions having particular usage in the study of analysis, physics, or another branch of science or mathematics. Typically, they come with their own conventional names and notations. Wolfram|Alpha has the ability to handle many families of special functions due to the powerful built-in functionalities of Wolfram Language.

Bessel & Related Functions ›

Compute properties for Bessel and Bessel-related functions.

Plot a Bessel function:

plot J2(x)

Integrate a spherical Bessel function:

integrate spherical Bessel j3(x)
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Elliptic Functions ›

Compute properties for elliptic functions and other related functions.

Plot an elliptic integral:

plot Elliptic E(t)

Compute the Taylor series of the elliptic nome:

series elliptic nome Q(m)

Calculate the Carlson function:

Carlson elliptic integral function Rf(1., 2., x)
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Legendre Polynomials

Compute properties for Legendre polynomials of the first and the second kind.

Compute Legendre P polynomials:

Legendre P(10,x)

Plot a Legendre Q polynomial:

plot Legendre Q(8,x)

Heun Functions

Visualize or compute properties for Heun functions.

Evaluate the z-derivative of the double-confluent Heun function:

evaluate heund'(1.2, -0.6, -0.7 , -0.18, 0.3, .12)

Plot values of the bi-confluent Heun function:

plot heunb(-.2, 1.3, 0.12, -0.14, x, -.3)

Plot a derivative of the general Heun function:

compute HeunG derivative [4.3 + 0.1 I, -0.2, 1.3, 0.12, -0.14, 4.32, 0.12]

Discover properties of the z-derivative of the tri-confluent Heun function:

HeunTPrime[x, -0.6, -0.7 , -0.18, 0.3 , 0.12]

Compute the Taylor series for values of the confluent Heun function:

series confluent heun function (4, -0.6, -0.7 , -0.18, 0.3 , x)

Beta Functions

Compute properties for the Euler beta or incomplete beta function.

Compute values of the Beta function:

Beta(5, 4)

Plot values of the incomplete Beta function:

plot Beta(x, 2, 3)

Error Functions

Compute properties for the error function and complementary error function.

Compute values of the error function:

erf(3)

Analyze the complementary error function:

integral representation erfc(z)

Spheroidal Functions

Compute properties for spheroidal functions of the first and the second kind.

Evaluate S(1) numerically:

Spheroidal S1(1,0,2.5,1)

Coulomb Functions

Visualize or compute properties for both outgoing and incoming irregular Coulomb functions.

Evaluate the incoming irregular Coulomb wave function at a point:

incoming irregular Coulomb wave 2, .2, 3.

Plot values of the Coulomb wave:

plot Coulomb wave of first type 3, 0.12, x

Analyze the derivative of an irregular Coulomb wave:

derivative irregular Coulomb wave 5, 0.46, u

RELATED EXAMPLES

  • Applied Mathematics
  • Calculus & Analysis
  • Complex Analysis
  • Differential Equations
  • Physics
    • Gamma & Related Functions ›

    Compute properties for gamma and gamma-related functions.

    Compute properties of the gamma function:

    Gamma(n)

    Simplify expressions involving gamma:

    Gamma(z)*Gamma(1-z)
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    Zeta Functions ›

    Compute properties for the family of zeta functions.

    Plot the zeta function on the critical line:

    plot zeta(1/2 + i*t)

    Numerically evaluate zeta near one of the nontrivial zeros:

    zeta(1/2 + 14.1347 i)
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