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Adrien-Marie Legendre introduced the polynomials named after him in 1782, defining them through their role in the multipole expansion in electrostatics.
Answer: True
Explanation: The source states that Adrien-Marie Legendre introduced the polynomials named after him in 1782, defining them through the generating function which is directly linked to the multipole expansion in electrostatics.
Legendre polynomials are defined as an orthogonal system over the interval [0,1] with a weight function of w(x)=1.
Answer: False
Explanation: Legendre polynomials are defined as an orthogonal system over the interval [-1,1] with a weight function of w(x)=1, not [0,1].
The standardization condition for Legendre polynomials requires that P_n(0)=1 for all n.
Answer: False
Explanation: The standardization condition for Legendre polynomials is P_n(1)=1, not P_n(0)=1.
P_0(x) = 1 and P_1(x) = x are the first two Legendre polynomials, derived from both the orthogonal system construction and the generating function expansion.
Answer: True
Explanation: The source confirms that P_0(x)=1 and P_1(x)=x are the first two Legendre polynomials, derivable from both the orthogonal system construction and the generating function expansion.
Legendre's differential equation is a first-order linear ordinary differential equation that has regular singular points at x = +/-1.
Answer: False
Explanation: Legendre's differential equation is a second-order linear ordinary differential equation, not first-order, and it does have regular singular points at x = +/-1.
Legendre polynomials are one of the three classical orthogonal polynomial systems, alongside Chebyshev and Jacobi polynomials.
Answer: False
Explanation: Legendre polynomials are one of the three classical orthogonal polynomial systems, alongside Laguerre and Hermite polynomials, not Chebyshev and Jacobi polynomials (though Chebyshev are related).
The generating function for Legendre polynomials is 1/sqrt(1+2xt+t^2).
Answer: False
Explanation: The generating function for Legendre polynomials is 1/sqrt(1-2xt+t^2), not 1/sqrt(1+2xt+t^2).
Legendre's differential equation has solutions that are always polynomials, regardless of the integer 'n'.
Answer: False
Explanation: While Legendre polynomials are polynomial solutions for integer 'n', Legendre's differential equation also has non-polynomial solutions known as Legendre functions of the second kind, Q_n.
After whom are Legendre polynomials named, and in what year were they introduced?
Answer: Adrien-Marie Legendre, 1782
Explanation: Legendre polynomials are named after Adrien-Marie Legendre, who introduced them in 1782.
When defined as an orthogonal system, what is the weight function for Legendre polynomials over the interval [-1,1]?
Answer: w(x) = 1
Explanation: When defined as an orthogonal system, Legendre polynomials use a weight function of w(x)=1 over the interval [-1,1].
What is the standardization condition applied to uniquely determine Legendre polynomials?
Answer: P_n(1) = 1
Explanation: The standardization condition applied to uniquely determine Legendre polynomials is P_n(1)=1.
What is the primary pedagogical advantage of defining Legendre polynomials via their construction as an orthogonal system?
Answer: It does not rely on the theory of differential equations and immediately demonstrates completeness.
Explanation: Defining Legendre polynomials via their construction as an orthogonal system is advantageous because it does not rely on differential equation theory and immediately demonstrates their completeness.
What is the generating function for Legendre polynomials?
Answer: 1/sqrt(1-2xt+t^2)
Explanation: The generating function for Legendre polynomials is 1/sqrt(1-2xt+t^2).
Legendre's differential equation is a second-order linear ordinary differential equation. What are its regular singular points?
Answer: x = +/-1
Explanation: Legendre's differential equation is a second-order linear ordinary differential equation with regular singular points at x = +/-1.
Bonnet's recursion formula is derived by differentiating the generating function with respect to 'x' and equating coefficients of powers of 't'.
Answer: False
Explanation: Bonnet's recursion formula is derived by differentiating the generating function with respect to 't', not 'x', and then equating coefficients of powers of 't'.
Rodrigues' formula provides a compact expression for Legendre polynomials, involving the nth derivative of (x^2 - 1)^n.
Answer: True
Explanation: Rodrigues' formula for Legendre polynomials is P_n(x) = (1 / (2^n * n!)) * (d^n / dx^n) * (x^2 - 1)^n, which involves the nth derivative of (x^2 - 1)^n.
The orthogonality and normalization of Legendre polynomials are expressed by an integral that equals (2 / (2n+1)) * delta_mn.
Answer: True
Explanation: The source states that the orthogonality and normalization of Legendre polynomials are expressed by the integral from -1 to 1 of P_m(x)P_n(x) dx = (2 / (2n+1)) * delta_mn.
The general recurrence relation for coefficients of a Legendre polynomial in a power series involves terms a_n,k and a_n,k-1.
Answer: False
Explanation: The general recurrence relation for coefficients of a Legendre polynomial in a power series involves terms a_n,k and a_n,k-2, not a_n,k-1.
The derivative of P_n+1(x) can be expressed as a sum of lower-degree Legendre polynomials with coefficients of the form (2k+1).
Answer: True
Explanation: The source states that the derivative of P_n+1(x) can be expressed as a sum of lower-degree Legendre polynomials, such as (2n+1)P_n(x) + (2(n-2)+1)P_n-2(x) + ..., which indeed have coefficients of the form (2k+1).
Sturm-Liouville theory demonstrates the orthogonality and completeness of Legendre polynomial solutions by rewriting Legendre's differential equation as an eigenvalue problem.
Answer: True
Explanation: The source states that Sturm-Liouville theory demonstrates the orthogonality and completeness of Legendre polynomial solutions by rewriting Legendre's differential equation as an eigenvalue problem.
The underivative formula for Legendre polynomials P_n(x) for n >= 1 is (1 / (2n+1)) * [P_n+1(x) + P_n-1(x)].
Answer: False
Explanation: The underivative formula for Legendre polynomials P_n(x) for n >= 1 is (1 / (2n+1)) * [P_n+1(x) - P_n-1(x)], not with a plus sign.
The derivative of a Legendre polynomial P_n(x) at the endpoint x=1 is given by P_n'(1) = n(n+1)/2.
Answer: True
Explanation: The source states that the derivative of a Legendre polynomial P_n(x) at the endpoint x=1 is given by P_n'(1) = n(n+1)/2.
Which of the following is Bonnet's recursion formula for Legendre polynomials?
Answer: (n+1)P_n+1(x) = (2n+1)xP_n(x) - nP_n-1(x)
Explanation: Bonnet's recursion formula for Legendre polynomials is (n+1)P_n+1(x) = (2n+1)xP_n(x) - nP_n-1(x).
What is Rodrigues' formula for Legendre polynomials?
Answer: P_n(x) = (1 / (2^n * n!)) * (d^n / dx^n) * (x^2 - 1)^n
Explanation: Rodrigues' formula for Legendre polynomials is P_n(x) = (1 / (2^n * n!)) * (d^n / dx^n) * (x^2 - 1)^n.
What does the Kronecker delta (delta_mn) represent in the combined orthogonality and normalization statement for Legendre polynomials?
Answer: It is 1 if m=n and 0 otherwise.
Explanation: The Kronecker delta (delta_mn) is 1 if m=n and 0 otherwise, indicating orthogonality for m != n and normalization for m = n.
According to the completeness property, how can a piecewise continuous function f(x) in [-1,1] be approximated by Legendre polynomials?
Answer: By a sequence of sums of Legendre polynomials.
Explanation: The completeness property states that any piecewise continuous function f(x) in [-1,1] can be approximated in the mean by a sequence of sums of Legendre polynomials.
What is the derivative of a Legendre polynomial P_n(x) at the endpoint x=1?
Answer: n(n+1)/2
Explanation: The source states that the derivative of a Legendre polynomial P_n(x) at the endpoint x=1 is given by P_n'(1) = n(n+1)/2.
Which of the following correctly describes the recurrence relation for coefficients a_n,k of a Legendre polynomial P_n(x) in a power series?
Answer: a_n,k = -((n-k+2)(n+k-1) / (k(k-1))) * a_n,k-2
Explanation: The recurrence relation for coefficients a_n,k of a Legendre polynomial P_n(x) in a power series is a_n,k = -((n-k+2)(n+k-1) / (k(k-1))) * a_n,k-2.
What is the recurrence relation that connects the derivative of P_n(x) to P_n(x) and P_n-1(x)?
Answer: ((x^2-1)/n) * (d/dx)P_n(x) = xP_n(x) - P_n-1(x)
Explanation: The recurrence relation that connects the derivative of P_n(x) to P_n(x) and P_n-1(x) is ((x^2-1)/n) * (d/dx)P_n(x) = xP_n(x) - P_n-1(x).
What is the underivative formula for Legendre polynomials P_n(x) for n >= 1?
Answer: integral P_n(x) dx = (1 / (2n+1)) * [P_n+1(x) - P_n-1(x)]
Explanation: The underivative formula for Legendre polynomials P_n(x) for n >= 1 is integral P_n(x) dx = (1 / (2n+1)) * [P_n+1(x) - P_n-1(x)].
The parity property of Legendre polynomials states that P_n(-x) = P_n(x) for all n, meaning they are always even functions.
Answer: False
Explanation: The parity property of Legendre polynomials is P_n(-x) = (-1)^n P_n(x), meaning they are even functions if 'n' is even and odd functions if 'n' is odd, not always even.
For any Legendre polynomial P_n(x) where n is greater than or equal to 1, its integral over the interval [-1,1] is 1.
Answer: False
Explanation: For any Legendre polynomial P_n(x) where n is greater than or equal to 1, its integral over the interval [-1,1] is 0, not 1.
The average of a function approximated by a Legendre series over [-1,1] is given by the leading expansion coefficient, a_0.
Answer: True
Explanation: The source states that when a function is approximated by a Legendre series over [-1,1], its average is given by the leading expansion coefficient, a_0.
The value of a Legendre polynomial P_n(x) at x=-1 is always 1, regardless of n.
Answer: False
Explanation: The value of a Legendre polynomial P_n(x) at x=-1 is (-1)^n, meaning it is 1 for even 'n' and -1 for odd 'n', not always 1.
All 'n' zeros of a Legendre polynomial P_n(x) are real, distinct, and located within the open interval (-1,1).
Answer: True
Explanation: The source confirms that all 'n' zeros of a Legendre polynomial P_n(x) are real, distinct, and lie within the open interval (-1,1).
The interlacing property of Legendre polynomial zeros states that each subinterval created by the zeros of P_n(x) contains exactly two zeros of P_n+1(x).
Answer: False
Explanation: The interlacing property states that each subinterval created by the zeros of P_n(x) contains exactly one zero of P_n+1(x), not two.
The Askey-Gasper inequality states that the sum of Legendre polynomials from j=0 to n of P_j(x) is less than or equal to 0 for x >= -1.
Answer: False
Explanation: The Askey-Gasper inequality states that the sum of Legendre polynomials from j=0 to n of P_j(x) is greater than or equal to 0 for x >= -1, not less than or equal to 0.
The zeros of Legendre polynomials are crucial in Gaussian quadrature for numerical integration.
Answer: True
Explanation: The source states that the zeros of Legendre polynomials play a crucial role in Gaussian quadrature for numerical integration.
Hilb's formula describes the asymptotic behavior of Legendre polynomials for small degrees.
Answer: False
Explanation: Hilb's formula describes the asymptotic behavior of Legendre polynomials as the degree 'l' approaches infinity, not for small degrees.
What is the parity property of Legendre polynomials P_n(x)?
Answer: P_n(-x) = (-1)^n P_n(x)
Explanation: The parity property of Legendre polynomials is P_n(-x) = (-1)^n P_n(x).
What is the integral of a Legendre polynomial P_n(x) over the interval [-1,1] for n >= 1?
Answer: 0
Explanation: For any Legendre polynomial P_n(x) where n is greater than or equal to 1, its integral over the interval [-1,1] is 0.
If a function is approximated by a Legendre series over [-1,1], what represents the average of that series?
Answer: The leading expansion coefficient, a_0
Explanation: The average of a function approximated by a Legendre series over [-1,1] is given by the leading expansion coefficient, a_0.
What is the value of P_n(-1) for a Legendre polynomial P_n(x)?
Answer: (-1)^n
Explanation: The value of P_n(-1) for a Legendre polynomial P_n(x) is (-1)^n.
What is the value of P_2n+1(0) for an odd-degree Legendre polynomial?
Answer: 0
Explanation: For an odd-degree Legendre polynomial P_2n+1(x), its value at the origin, P_2n+1(0), is 0.
What does the Askey-Gasper inequality state about the sum of Legendre polynomials from j=0 to n of P_j(x) for x >= -1?
Answer: The sum is greater than or equal to 0.
Explanation: The Askey-Gasper inequality states that the sum of Legendre polynomials from j=0 to n of P_j(x) is greater than or equal to 0 for x >= -1.
What property do all 'n' zeros of a Legendre polynomial P_n(x) possess?
Answer: They are all real, distinct, and lie within the open interval (-1,1).
Explanation: All 'n' zeros of a Legendre polynomial P_n(x) are real, distinct, and lie within the open interval (-1,1).
What is the significance of Legendre polynomial zeros in numerical integration?
Answer: They are crucial for Gauss-Legendre quadrature.
Explanation: The zeros of Legendre polynomials are crucial for Gauss-Legendre quadrature, a method used in numerical integration.
What does Hilb's formula describe for Legendre polynomials?
Answer: Their asymptotic behavior as the degree 'l' approaches infinity.
Explanation: Hilb's formula describes the asymptotic behavior of Legendre polynomials as the degree 'l' approaches infinity.
Legendre polynomials were initially applied in physics to describe the electric potential due to a continuous charge distribution.
Answer: True
Explanation: Legendre polynomials were initially applied in physics as coefficients in the expansion of the Newtonian potential, which describes gravitational or Coulomb potential due to a point mass or charge, useful for integrating over continuous distributions.
In solving Laplace's equation in spherical coordinates with axial symmetry, the potential is expressed as a sum involving Legendre polynomials of cos(theta).
Answer: True
Explanation: The source indicates that when solving Laplace's equation in spherical coordinates with axial symmetry, the potential Phi(r,theta) is expressed as a sum involving Legendre polynomials of cos(theta).
Legendre polynomials are used in recurrent neural networks to optimize networks containing a d-dimensional memory vector by approximating the input's sliding window with shifted Legendre polynomials.
Answer: True
Explanation: The source states that Legendre polynomials are used in recurrent neural networks to optimize networks by approximating the input's sliding window with a linear combination of shifted Legendre polynomials.
Legendre polynomials of a scalar product of unit vectors can be expanded using spherical harmonics, involving a sum from m=-l to l.
Answer: True
Explanation: The source states that Legendre polynomials of a scalar product of unit vectors can be expanded using spherical harmonics, involving a sum from m=-l to l.
Legendre polynomials were first introduced in physics as coefficients in the expansion of what type of potential?
Answer: The Newtonian potential
Explanation: Legendre polynomials were first introduced in physics as coefficients in the expansion of the Newtonian potential.
In the multipole expansion for electric potential due to a point charge on the z-axis at z=a, if the observation point's radius 'r' is greater than 'a', how is the potential expanded?
Answer: Phi(r,theta) is proportional to (1/r) * sum (a/r)^k * P_k(cos theta)
Explanation: If r > a, the electric potential is expanded as Phi(r,theta) is proportional to (1/r) * sum (a/r)^k * P_k(cos theta).
In recurrent neural networks, what is approximated by a linear combination of shifted Legendre polynomials to optimize networks?
Answer: The sliding window of the input
Explanation: In recurrent neural networks, the sliding window of the input is approximated by a linear combination of shifted Legendre polynomials to optimize networks.