Homework Answers
2 (Dirac Delta Function) Using properties of the delta function and the relations discussed in class,...
2 (Dirac Delta Function) Using properties of the delta function and the relations discussed in class, <xx'> = 8(x – z'), ><x=1 Prove that | (x – y)8(y – 2)f(z)dydz = f(x).
The fourier Transform of a dirac delta function, 8(t) is: (a) X(f) = 11-20,00)(f) (b) X(f)...
The fourier Transform of a dirac delta function, 8(t) is: (a) X(f) = 11-20,00)(f) (b) X(f) = 8(f) (c) X(f) = 0 (d) None of the Above
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined...
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined by δ(z) = 0 if |メ0, δ(z) = oo ifx-0 but the integral of the function over any interval containing the zero of the argument is unity, Equivalent, if f(x) is continuous at the origin One should treat the Dirac δ function as a limit of a sequence of functions peaked at x-0 As the limit is approached, the height of the peak increases...
(1/x) does not exist but that lim cos(1/x) = 0 a) Prove that lim cos x-+0...
(1/x) does not exist but that lim cos(1/x) = 0 a) Prove that lim cos x-+0
(1/x) does not exist but that lim cos(1/x) = 0 a) Prove that lim cos x-+0
(i) Write down the mathematical definition of the Dirac delta function. (ii) Compute the following integrals...
(i) Write down the mathematical definition of the Dirac delta function. (ii) Compute the following integrals | ** cos(e)8(t – 2) dt f sin®(t + 7/3)8'(t) dt e '8(t? – r) dt where x is a real non-zero number.
Let f(x)={user user = { x 8. Prove the following 10 a. Prove lim f(x) =...
Let f(x)={user user = { x 8. Prove the following 10 a. Prove lim f(x) = 0 b. Prove lim f(x)=1 c. Prove lim f(x) does not exist. 1-2
Let f(x) be a differentiable function with inverse of f(x) such that f(0)=0 and f'(0) is not 0. Prove lim(x->0) f...
Let f(x) be a differentiable function with inverse of f(x) such that f(0)=0 and f'(0) is not 0. Prove lim(x->0) f(x)/f −1(x) =f'(0)^2 f-1(x) is f inverse of x
XL Xa 12. (a) Suppose that f(x) = g(x) for all x. Prove that lim f(x)...
XL Xa 12. (a) Suppose that f(x) = g(x) for all x. Prove that lim f(x) < lim g(x), provided that these limits exist. (b) How can the hypotheses be weakened? (c) If f(x) < g(x) for all x, does it necessarily follow that lim f (x) < lim g(x)? Ya X-
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.) (c) Let f :la,b- R be an integrable function. Prove tha...
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.)
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) =
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
2 (Dirac Delta Function) Using properties of the delta function and the relations discussed in class, <xx'> = 8(x – z'), ><x=1 Prove that | (x – y)8(y – 2)f(z)dydz = f(x).
The fourier Transform of a dirac delta function, 8(t) is: (a) X(f) = 11-20,00)(f) (b) X(f) = 8(f) (c) X(f) = 0 (d) None of the Above
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined by δ(z) = 0 if |メ0, δ(z) = oo ifx-0 but the integral of the function over any interval containing the zero of the argument is unity, Equivalent, if f(x) is continuous at the origin One should treat the Dirac δ function as a limit of a sequence of functions peaked at x-0 As the limit is approached, the height of the peak increases...
(1/x) does not exist but that lim cos(1/x) = 0 a) Prove that lim cos x-+0
(1/x) does not exist but that lim cos(1/x) = 0 a) Prove that lim cos x-+0
(i) Write down the mathematical definition of the Dirac delta function. (ii) Compute the following integrals | ** cos(e)8(t – 2) dt f sin®(t + 7/3)8'(t) dt e '8(t? – r) dt where x is a real non-zero number.
Let f(x)={user user = { x 8. Prove the following 10 a. Prove lim f(x) = 0 b. Prove lim f(x)=1 c. Prove lim f(x) does not exist. 1-2
XL Xa 12. (a) Suppose that f(x) = g(x) for all x. Prove that lim f(x) < lim g(x), provided that these limits exist. (b) How can the hypotheses be weakened? (c) If f(x) < g(x) for all x, does it necessarily follow that lim f (x) < lim g(x)? Ya X-
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.)
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) =
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
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