Legendre Equation and Polynomials Quiz

The Legendre polynomials are a set of polynomials solutions to which differential equation?

• Euler equation

• Laguerre equation

• Airy equation

• Hermite equation

The Legendre equation has regular singular points at x=±1.

• True

• False

The Legendre equation is a special case of which differential equation?

• Laplace equation

• Bessel equation

• Hermite equation

• Legendre equation

The Legendre equation is used to solve problems in which area of mathematics?

• Number theory

• Algebra

• Differential equations

• Trigonometry

The Legendre equation arises in the study of what physical phenomena?

• Quantum mechanics

• Electromagnetism

• Fluid dynamics

• General relativity

The Legendre polynomials are orthogonal with respect to what weight function?

• e^(-x)

• e^(-x^2)

• x^2

• 1

What is the generating function for the Legendre polynomials?

• e^(xt/(t^2-1))

• e^(tx/(t^2-1))

• e^(-xt/(t^2-1))

• e^(-tx/(t^2-1))

What is the recurrence relation for the Legendre polynomials?

• P_n(x)=(2n-1)xP_(n-1)(x)-(n-1)P_(n-2)(x)

• P_n(x)=(2n+1)xP_(n-1)(x)-(n+1)P_(n-2)(x)

• P_n(x)=(n+1)xP_(n-1)(x)-nP_(n-2)(x)

• P_n(x)=(n-1)xP_(n-1)(x)-(n-2)P_(n-2)(x)

The Legendre polynomials form a complete set in what interval?

• [0,1]

• [-1,1]

• [1,∞)

• (-∞,∞)

What is the derivative of the Legendre polynomial of order n, P_n(x)?

• P_n'(x)=P_n(x+1)

• P_n'(x)=nP_n-1(x)

• P_n'(x)=P_n(x-1)

• P_n'(x)=P_n+1(x)