What are the singular points of Legendre differential equation?

What are the singular points of Legendre differential equation?

Legendre Equation: The points x = ±1 are singular points, since P(x) = 1- x2 is zero there. All other points are ordinary points.

Why do we use Series solutions?

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

Why do we use Legendre equations?

For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom [3], [4] and in the determination of potential functions in the spherically symmetric geometry [5], etc.

What is the degree of Legendre equation?

Legendre’s differential equation has the form (1 − x2)y − 2xy + l(l + 1)y = 0, (2) where the parameter l, which is a real number, (we take l = 0,1,2,ททท), is called the degree.

What is ordinary and singular point?

In mathematics, in the theory of ordinary differential equations in the complex plane , the points of. are classified into ordinary points, at which the equation’s coefficients are analytic functions, and singular points, at which some coefficient has a singularity.

How do you find the Indicial equation?

y′=∞∑k=0(k+r)akxk+r−1,y″=∞∑k=0(k+r)(k+r−1)akxk+r−2. 4r(r−1)+1=0. This equation is called the indicial equation. This particular indicial equation has a double root at r=12.

How to get a series solution to the Legendre equation?

If you start with the Legendre equation and differentiate it $m$ times, you end up with a new differential equation for the functions $y=P_n^{(m)}$ of the form $$ (1-x^2)y”-2(m+1)xy’+(n(n+1)-m(m+1))y = 0. $$ It is this form of the associated Legendre equation where you can get a series solution.

When is the Legendre ordinary differential equation encountered?

In that case the parameters are usually labelled with Greek letters. The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace’s equation (and related partial differential equations) in spherical coordinates.

When do Legendre polynomials have nonsingular solutions?

This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials.

Which is the canonical solution of the general Legendre equation?

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation. or equivalently. where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively.