Here are some notes on special functions and where they may arise. We consider functions in applied mathematics beyond field (four arithmetic operations), composition and inverse operations applied to the power and exponential functions.

#### 1. Bessel and related functions

Bessel functions of the first () and second () kind of order satisfy:

.

Solutions for integer arise in solving Laplace’s equation in cylindrical coordinates while solutions for half-integer arise in solving the Helmholtz equation in spherical coordinates. Hence they come about in wave propagation, heat diffusion and electrostatic potential problems. The functions oscillate roughly periodically with amplitude decaying proportional to . Note that is the second linearly independent solution when is an integer (for integer , ). Also, for integer , has the generating function

the integral representations

and satisfies the orthogonality relation

where , Kronecker delta, and is the m-th zero of .

Modified Bessel functions of the first () and second () kind of order satisfy:

(replacing with in the previous equation).

The four functions may be expressed as follows.

(In the last formula we need to take a limit when is an integer.)

Note that and are singular at zero.

The Hankel functions and are also known as Bessel functions of the third kind.

The functions , , , and all satisfy the recurrence relations (using in place of any of these four functions)

Bessel functions of higher orders/derivatives can be calculated from lower ones via:

In particular, note that is the derivative of .

The Airy functions of the first () and second () kind satisfy

.

This arises as a solution to Schrödinger’s equation for a particle in a triangular potential well and also describes interference and refraction patterns.

#### 2. Orthogonal polynomials

Hermite polynomials (the probabilists’ defintion) can be defined by:

,

and are orthogonal with respect to weighting function on .

They satisfy the differential equation

(where is forced to be an integer if we insist be polynomially bounded at )

and the recurrence relation

.

The first few such polynomials are . The Physicists’ Hermite polynomials are related by and arise for example as the eigenstates of the quantum harmonic oscillator.

Laguerre polynomials are defined by

,

and are orthogonal with respect to on .

They satisfy the differential equation

,

recurrence relation

,

and have generating function

The first few values are . Note also that .

The functions come up as the radial part of solution to Schrödinger’s equation for a one-electron atom.

Legendre polynomials can be defined by

and are orthogonal with respect to the norm on .

They satisfy the differential equation

,

recurrence relation

and have generating function

.

The first few values are .

They arise in the expansion of the Newtonian potential (multipole expansions) and Laplace’s equation where there is axial symmetry (spherical harmonics are expressed in terms of these).

Chebyshev polynomials of the *1st kind* can be defined by

and are orthogonal with respect to weighting function in .

They satisfy the differential equation

,

the relations

and have generating function

The first few values are . These polynomials arise in approximation theory, namely their roots are used as nodes in piecewise polynomial interpolation. The function is the polynomial of leading coefficient 1 and degree n where the maximal absolute value on (-1,1) is minimal.

Chebyshev polynomials of the *2nd kind* are defined by

and are orthogonal with respect to weighting function in .

They satisfy the differential equation

,

the recurrence relation

and have generating function

The first few values are . (There are also less well known Chebyshev polynomials of the third and fourth kind.)

Bessel polynomials may be defined from Bessel functions via

.

They satisfies the differential equation

.

The first few values are .

**3. Integrals**

The error function has the form

.

This can be interpreted as the probability a normally distributed random variable with zero mean and variance 1/2 is in the interval .

The cdf of the normal distribution $\Phi(x)$ is related to this via . Hence the tail probability of the standard normal distribution is .

Fresnel integrals are defined by

They have applications in optics.

The exponential integral (used in heat transfer applications) is defined by

.

It is related to the logarithmic integral

by (for real ).

The incomplete elliptic integral of the first, second and third kinds are defined by

Setting gives the complete elliptic integrals.

Any integral of the form , where is a constant, is a rational function of its arguments and is a polynomial of 3rd or 4th degree with no repeated roots, may be expressed in terms of the elliptic integrals. The circumference of an ellipse of semi-major axis , semi-minor axis and eccentricity is given by , where is the complete integral of the second kind.

(Some elliptic functions are related to inverse elliptic integral, hence their name.)

The (upper) incomplete Gamma function is defined by

.

It satisfies the recurrence relation . Setting gives the Gamma function which interpolates the factorial function.

The digamma function is the logarithmic derivative of the gamma function:

.

Due the relation , this function appears in the regularisation of divergent integrals, e.g.

.

The incomplete Beta function is defined by

.

When setting this becomes the Beta function which is related to the gamma function via

.

This can be extended to the multivariate Beta function, used in defining the Dirichlet function.

.

The polylogarithm, appearing as integrals of the Fermi–Dirac and Bose–Einstein distributions, is defined by

Note the special case and the case is known as the dilogarithm. We also have the recursive formula

.

#### 4. Generalised Hypergeometric functions

All the above functions can be written in terms of generalised hypergeometric functions.

where for or .

The special case is called a confluent hypergeometric function of the first kind, also written .

This satisfies the differential equation (Kummer’s equation)

.

The Bessel, Hankel, Airy, Laguerre, error, exponential and logarithmic integral functions can be expressed in terms of this.

The case is sometimes called Gauss’s hypergeometric functions, or simply hypergeometric functions. This satisfies the differential equation

.

The Legendre, Hermite and Chebyshev, Beta, Gamma functions can be expressed in terms of this.

#### Further reading

The Wolfram Functions Site

Wikipedia: List of mathematical functions

Wikipedia: List of special functions and eponyms

Wikipedia: List of q-analogs

Wikipedia Category: Orthogonal polynomials

Weisstein, Eric W. “Laplace’s Equation.” From *MathWorld*–A Wolfram Web Resource. http://mathworld.wolfram.com/LaplacesEquation.html

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