Unit 3: Special Functions & Partial Differential Equations

1. Properties of Legendre Polynomials

Legendre polynomials, denoted as Pn(x), are a set of orthogonal polynomials that emerge as solutions to the Legendre differential equation. In physics, they are particularly important in problems involving spherical symmetry, such as gravitation and electrostatics.

Rodrigues Formula

This formula provides a systematic way to generate any Legendre polynomial by differentiating a specific function n times.

Generating Function

The entire set of Legendre polynomials can be generated from the power series expansion of the following function:

Orthogonality

Legendre polynomials are orthogonal over the interval [-1, 1]. This means the integral of the product of two different Legendre polynomials is zero.

Integral from -1 to 1 of [Pm(x) * Pn(x)] dx = 0, if m is not equal to n.

Simple Recurrence Relations

These relations connect polynomials of different orders, which is useful for simplifying complex expressions:

• (n + 1)Pn+1(x) = (2n + 1)xPn(x) - nPn-1(x)
• nPn(x) = xPn'(x) - Pn-1'(x)

2. Expansion in Legendre Series

Any piecewise continuous function f(x) defined in the interval [-1, 1] can be expressed as a series of Legendre polynomials:

where the coefficient An is determined by using the orthogonality property:

3. Bessel Functions of the First Kind

Bessel functions, denoted as Jn(x), arise in problems involving cylindrical or circular symmetry. They are solutions to the Bessel differential equation.

Generating Function for Jn(x)

Bessel functions of the first kind can be obtained from the following expansion:

Recurrence Relations for Jn(x)

Common relations used to calculate higher-order Bessel functions from lower-order ones:

• 2n Jn(x) = x [Jn-1(x) + Jn+1(x)]
• 2 Jn'(x) = Jn-1(x) - Jn+1(x)

4. Zeros and Orthogonality of Bessel Functions

Zeros of J0(x) and J1(x)

The values of x for which Jn(x) = 0 are called the zeros of the Bessel function. These zeros are crucial in physics for determining vibrational modes of circular membranes or electromagnetic fields in cylindrical cavities.

Orthogonality Property

Bessel functions of the same order n but with different arguments scaled by zeros are orthogonal.

Integral from 0 to a of [x * Jn(αm * x / a) * Jn(αk * x / a)] dx = 0, if m is not equal to k.

5. Partial Differential Equations: Separation of Variables

The method of separation of variables is a powerful technique to solve partial differential equations (PDEs) by assuming the solution is a product of functions, each depending on only one variable.

The Steps:

1. Assume a product solution: e.g., Ψ(x, y) = X(x) * Y(y).
2. Substitute this into the PDE.
3. Divide the equation such that variables are separated on different sides.
4. Equate each side to a separation constant.
5. Solve the resulting ordinary differential equations (ODEs).

6. Laplace's Equation in Rectangular Symmetry

Laplace's equation is written as:

In 2D rectangular coordinates (x, y), it is:

Using separation of variables Ψ(x, y) = X(x)Y(y), we obtain:

• d^2 X / dx^2 = -k^2 * X
• d^2 Y / dy^2 = k^2 * Y

The general solution depends on the boundary conditions of the physical problem, often involving sines, cosines, and hyperbolic functions.