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2. Consider the set of functions {f(x)} of the real variable x, defined on the interval...
2) The set S of all real-valued functions f(x) of a single real variable z is...
2) The set S of all real-valued functions f(x) of a single real variable z is a vector space. (a) Show that the set L of all real-valued linear functions f(x) = mx + b of a single variable x is a subspace of S. (b) Show tha (f(x), g(x))= | f(z)g(x)dx is an inner product on L. (c) Find an orthonormal basis for C with respect to the inner product defined in (b)
are defined on the interval a <t< b #14. Assume that the real functions f(t) and...
are defined on the interval a <t< b #14. Assume that the real functions f(t) and p(t) that f is positive and continuous and p is integrable. Prove that f(t)e'dt< f(t)dt, a. a and that equality holds if and only if the function p assumes the same value mod 27r in all its points of continuity
are defined on the interval a
2. For the following problems, indicate if the statement is TRUE or FALSE. If the statement...
2. For the following problems, indicate if the statement is TRUE or FALSE. If the statement is FALSE, state why and make necessary corrections. [13 points total] 1. If two wave functions are orthogonal, then they are also both normalized. b. If a system is in a state described by a wavefunction Y, then the average value of the observable corresponding to the operator A is given by the equation YAY dt (a) [ 'ዋ dr c. The momentum operator,...
# 4: For smooth complex valued functions f(x), g(z) defined for 0 < x inner product<f(x),g(x) > by 2π define the Hermitian Introduce the operator D(f() a)Show that <D(f(x),9()), D(g(x)) &...
# 4: For smooth complex valued functions f(x), g(z) defined for 0 < x inner product<f(x),g(x) > by 2π define the Hermitian Introduce the operator D(f() a)Show that <D(f(x),9()), D(g(x)) > if f b) For n and integer show that einz for 0-x-2n satisfi c) Show that for mメn both integers then < einz, enny-0, 0,警) (0)- ic boundary conditions. Also onormal and < einz, einz >-2T. θ, Call these last periodic boundary conditions for f(x), g(s), show that D(einz)...
1. Consider the function defined by f(x) 0, |x| < 2 1 and f(x) f(x 4) (a) Sketch the graph of f(x) on the interval -...
1. Consider the function defined by f(x) 0, |x| < 2 1 and f(x) f(x 4) (a) Sketch the graph of f(x) on the interval -6,6 8 (b) Find the Fourier series representation of f(z). You must show how to evaluate any integrals that are needed
1. Consider the function defined by f(x) 0, |x|
4. Consider the sequence of functions fu(x) = 1 timer defined on the whole real line...
4. Consider the sequence of functions fu(x) = 1 timer defined on the whole real line R. f pointwise as n → for (a) (5 pts.) Find the function f such that o DE-1,1) (b) (5 pts.) Find max: (2) -f(x). Is the convergence in (a) uniform?
1. Show clearly whether each is, or is not, a directed set: (a) The real interval...
1. Show clearly whether each is, or is not, a directed set: (a) The real interval (0,0) with a b defined to mean a <b. (b) The set of all finite subsets of Z where S T means that S and T are disjoint and S has more elements than T. → R with f g defined to mean (c) The collection of all functions f:R f(1) 9(0) Definition 3.2.1. A relation > on a nonempty set X is a...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For a...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For any number ce [0, 1] (a) Show that the set R is a ring and that the set Ic is an ideal of R. (b) Is I UI2 and ideal? Is I, nI an ideal?
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval...
10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c)...
From Arfken
10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c) a weighting function w(x)-xe. Use the Gram-Schmidt procedure to construct the first three orthonormal functions from the set un(x) for this interval and this weighting function.
10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c) a weighting function w(x)-xe. Use the Gram-Schmidt procedure...
For the real-valued functions f (x)=x +4 and g(x)=x-2, find the composition f ºg and specify...
For the real-valued functions f (x)=x +4 and g(x)=x-2, find the composition f ºg and specify its domain using interval notation. 昌 石 (reg)(x) = 0 可回(口,口口,口 OUD (口,可 口,口) Domain of fog:口 0 x 0 -00 6 ?
2) The set S of all real-valued functions f(x) of a single real variable z is a vector space. (a) Show that the set L of all real-valued linear functions f(x) = mx + b of a single variable x is a subspace of S. (b) Show tha (f(x), g(x))= | f(z)g(x)dx is an inner product on L. (c) Find an orthonormal basis for C with respect to the inner product defined in (b)
are defined on the interval a <t< b #14. Assume that the real functions f(t) and p(t) that f is positive and continuous and p is integrable. Prove that f(t)e'dt< f(t)dt, a. a and that equality holds if and only if the function p assumes the same value mod 27r in all its points of continuity
are defined on the interval a
2. For the following problems, indicate if the statement is TRUE or FALSE. If the statement is FALSE, state why and make necessary corrections. [13 points total] 1. If two wave functions are orthogonal, then they are also both normalized. b. If a system is in a state described by a wavefunction Y, then the average value of the observable corresponding to the operator A is given by the equation YAY dt (a) [ 'ዋ dr c. The momentum operator,...
# 4: For smooth complex valued functions f(x), g(z) defined for 0 < x inner product<f(x),g(x) > by 2π define the Hermitian Introduce the operator D(f() a)Show that <D(f(x),9()), D(g(x)) > if f b) For n and integer show that einz for 0-x-2n satisfi c) Show that for mメn both integers then < einz, enny-0, 0,警) (0)- ic boundary conditions. Also onormal and < einz, einz >-2T. θ, Call these last periodic boundary conditions for f(x), g(s), show that D(einz)...
1. Consider the function defined by f(x) 0, |x| < 2 1 and f(x) f(x 4) (a) Sketch the graph of f(x) on the interval -6,6 8 (b) Find the Fourier series representation of f(z). You must show how to evaluate any integrals that are needed
1. Consider the function defined by f(x) 0, |x|
4. Consider the sequence of functions fu(x) = 1 timer defined on the whole real line R. f pointwise as n → for (a) (5 pts.) Find the function f such that o DE-1,1) (b) (5 pts.) Find max: (2) -f(x). Is the convergence in (a) uniform?
1. Show clearly whether each is, or is not, a directed set: (a) The real interval (0,0) with a b defined to mean a <b. (b) The set of all finite subsets of Z where S T means that S and T are disjoint and S has more elements than T. → R with f g defined to mean (c) The collection of all functions f:R f(1) 9(0) Definition 3.2.1. A relation > on a nonempty set X is a...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For any number ce [0, 1] (a) Show that the set R is a ring and that the set Ic is an ideal of R. (b) Is I UI2 and ideal? Is I, nI an ideal?
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval...
From Arfken
10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c) a weighting function w(x)-xe. Use the Gram-Schmidt procedure to construct the first three orthonormal functions from the set un(x) for this interval and this weighting function.
10.3.4 You are given (a) a set of functions un (x)--x", n = 0, 1, 2, (b) an interval (0, oo), (c) a weighting function w(x)-xe. Use the Gram-Schmidt procedure...
For the real-valued functions f (x)=x +4 and g(x)=x-2, find the composition f ºg and specify its domain using interval notation. 昌 石 (reg)(x) = 0 可回(口,口口,口 OUD (口,可 口,口) Domain of fog:口 0 x 0 -00 6 ?
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