Special Functions

What is Special Function?

• Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

• There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special.

(Ref. Wikipedia)

Differential Equation

• A DE is a mathematical equation that relates some functions with its derivatives.

In Applications,

functions usually represent physical quantities and

the derivatives represent the rate of change, and the

equation defines the relation between the two.

Examples of differential Eqns.Radioactivity

Simple harmonic motion

Other examples, wave equation, Heat Equation, etc..

Types of differential equations

Ordinary differential equations (ODE) Partial differential equations (PDE)

Linear Non-Linear (Complex to solve)

(Simple to solve)

To be more general, the general form of an order n linear differential equation is

𝒂𝟎 (x)𝒅𝒏𝒚

𝒅𝒙𝒏+ 𝒂𝟏 (x)

𝒅𝒏−𝟏𝒚

𝒅𝒙𝒏−𝟏+……. 𝒂𝒏−𝟏 (x)

𝒅 𝒚

𝒅𝒙+ 𝒂𝒏 (x)y = f(x)

a0,a1, …. an and f are given functions of independent variable x and a0 !=0 .This equation is linear and y is derivatives

If n =2 𝑑2𝑦

𝑑𝑥2+ P(x)

𝑑 𝑦

𝑑𝑥+ 𝑄 (x)y = f(x)

The Bessels’ Differential equation is given by

𝒙𝟐𝒅𝟐𝒚

𝒅𝒙𝟐+ 𝒙

𝒅𝒚

𝒅𝒙+ 𝒙𝟐 −𝒏𝟐 𝒚 = 𝟎

or 𝒅𝟐𝒚

𝒅𝒙𝟐+

𝟏

𝒙

𝒅𝒚

𝒅𝒙+ 𝟏 −

𝒏𝟐

𝒙𝟐𝒚 = 𝟎 …………(1)

In which ‘n′ is a constant, called order of the Equation.

The solution of the equations are known as Bessel’s Functions and it has enormous use in different branches of mathematical physics and Engineering.

Solution to the Bessel equationsConcrete representation of the general solution depends on thenumber n.

Further we consider separately two cases:

• The order n is non-integer;

• The order n is an integer.

Case 1. The Order n is an not an Integer

Assuming that the number n is non-integer and positive, the general solution of the Bessel equation can be written as

y(x)= A Jn(x) + B J−n(x),

where A, B are arbitrary constants and Jn(x), J−n(x) are Bessel functions of the first kind.

Estimate the values of a0,a1,a2,….. an and derive the relation for Jn(x)

………

……..

The two solutions of 𝐽𝑛(𝑥) and 𝐽−𝑛(𝑥) [𝑛 is not an integer] is given by

𝑱𝒏 𝒙 = 𝒓=𝟎∞

(−𝟏)𝒓𝒙

𝟐

𝒏+𝟐𝒓

𝒓 ! 𝜞(𝒏+𝒓+𝟏)……….…….(2)

&

𝑱−𝒏 𝒙 = 𝒓=𝒏∞

(−𝟏)𝒓𝒙

𝟐

𝟐𝒓−𝒏

𝒓 ! 𝜞(𝒓−𝒏+𝟏)……….…(3)

• Put 𝑟 = 𝑠 + 𝑛 in equation (3), we get

𝑱−𝒏 𝒙 = 𝒓=𝒏∞

(−𝟏)𝒓𝒙

𝟐

−𝒏+𝟐𝒓

𝒓 ! 𝜞(−𝒏+𝒓+𝟏)……….…(3)

For non-integer n, the functions Jn(x) and J−n(x) are linearly independent, and are

therefore the two solutions of the differential equation.

On the other hand, for integer order n, the following relationship is valid

This means that the two solutions are no longer linearly independent. In this case,

the second linearly independent solution is then found to be the Bessel function of

the second kind.

Bessel’s Function Examples

Another example

• Bessel Equation is something which often occurs in Engineering and Physics which deals

with the Cylindrical Coordinates such as Circular Plates, Circular Membranes etc.

• Bessel Equation actually tried to deal with the singularities happening most of the times.

• Neuman problems, Vibration of Circular Membranes, Heating Equation in plates

has a good deal of use of Bessel Equations in them.

Some practical situations where Bessel form is used are surface vessel design (maritime

engineering), distribution of heat over an area (say, in smartphones), pressure vessel

design, microphone design and many others.

Generating FunctionsFor a sequence, a formal power series whose coefficients are the members of that sequence.

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series. This formal power series is the generating function

Recurrance Relation

• A mathematical relationship expressing the members of a sequence as

some combination of their predecessors

• Mathematical relationship expressing fn as some combination of fi

with i<n. When formulated as an equation to be solved, recurrence

relations are known as recurrence equations, or sometimes

difference equations. (Check the NOTES LINK for details)

A