Examples for

Bessel & Related Functions

Bessel functions are defined as solutions of a second-order differential equation, namely Bessel's differential equation. Commonly, there are two classes of solutions, called Bessel functions of the first kind and Bessel functions of the second kind. The linear combinations of these two are usually called Bessel functions of the third kind, or Hankel functions. Wolfram|Alpha has the ability to compute properties for the family of Bessel functions, as well as other Bessel-related functions, such as Airy and Struve functions.
Bessel Functions

Compute properties for different kinds of Bessel functions.

Plot a Bessel function of the first kind:

plot J2(x)

Integrate a spherical Bessel function:

integrate spherical Bessel j3(x)

Find differential equations satisfied by a Bessel function of the second kind:

differential equations Y_2(x)
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Airy Functions

Compute properties for four varieties of Airy functions.

Compute a value of an Airy function to high precision:

Ai(3) to 100 places

Plot an Airy function:

plot Airy Bi(x)

Find relations between functions:

relations between Gi(z) and Ai(z)

Find differential equations satisfied by an Airy function:

differential equation Gi(z)
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Struve Functions

Compute properties for the Struve H function and modified Struve L function.

Find information about the Struve functions:

StruveH(1, x)
StruveL(x, x)
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Hankel Functions

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

Generate information about the Hankel functions:

Hankel H1
Hankel H2

Compute information about the spherical Hankel functions:

SphericalHankelH1[2, z]
SphericalHankelH2[1, 3+i] to 50 digits
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