(i) Write down the mathematical definition of the Dirac delta function. (ii) Compute the following integrals...
answer 1,2,3,4 thank you.
HW4.5: Problem 1 Previous Problem Problem List Next Problem 1 point) Evaluate each of the integrals (here &(t) is the Dirac delta function) (60-3)dt (2)cos(3t)S(t -2) dt- (3)/eTst cos(4t)(t - 3) dt - c0 sin()(t - 5) dt-
HW4.5: Problem 1 Previous Problem Problem List Next Problem 1 point) Evaluate each of the integrals (here &(t) is the Dirac delta function) (60-3)dt (2)cos(3t)S(t -2) dt- (3)/eTst cos(4t)(t - 3) dt - c0 sin()(t - 5) dt-
6. In this problem you will learn how to use Dirac delta functions to solve integrals and define densities of point charges. (a) Using the definition of Dirac delta function, evaluate the following integrals 15) 产00 (i) (4x2-8x-1) δ(x-4) dx (ii) sin x δ(x-π/2) dx x3 δ(x + 3)dx In(x + 3)δ(x + 2)dx (b) What is the volume charge density of an electric dipole, consisting of a point charge -q at (c) What is the integral of this charge...
Page 2 II. (7) Use the Laplace transform to solve the IVP y" - 5y' + 6y = 8(t-1), y(0) = 0,0) = 0, where the right hand side is the Dirac Delta Function (t - to) for to = 1. You may use the partial fraction decomposition 1 + 52-58 +6 2 S-3 but you need to show all the steps needed to arrive to the expression 1 52-58 +6 in order to receive credit. f(t)=L-'{F(s) Table of Laplace...
please simplify
Problem 2.3 Evaluate or simplify the following integrals or expression as much as possible (show your work). (a) L, 8(t)x(t – 1)dt (e) , 8(at)dt (i) cos(10zt) [8(t) + 8(t + 5)] sin (b) 8(t – T)x(t)dt (f) 8(2t – 5) sin nt dt (c) L 8(t)x(r – t)dt cos (x - 5)|6(x – 3)dx (sin ke (B) e*-2 8(w) (k) 6(r – t)x(t)dt (d) (h) Jt-11 t+9 8(1 – 3)đr Problem 2.3 Evaluate or simplify the following...
8. (i) Find C[F(t)], where F(t) = { if 0 st 34, ift> 4 (ii) Compute the convolution e2 et directly by the definition of the convolution (iii) Evaluate Lle-2445 - e cos(4t) + sin(V2t)). blom.
Evaluate the following integrals (from A to E) A. Integration by parts i) ſ (3+ ++2) sin(2t) dt ii) Z dz un (ricos x?cos 4x dx wja iv) (2 + 5x)eš dr. B. Involving Trigonometric functions 271 п i) | sin? ({x)cos*(xx) dx ii) Sco -> (=w) sins (įw) iii) sec iv) ſ tan” (63)sec^® (6x) dx . sec" (3y)tan?(3y)dy C. Involving Partial fractions 4 z? + 2z + 3 1) $77 dx 10 S2-6922+4) dz x2 + 5x -...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite square well -a < x < a is given by 2 sin2 V inside the well and zero outside. Find the probability that a measurement of energy yields the eigenvalue En. (Hint: use data on page 6.) [6] Useful Data and Formulas = 1.60 x 10-19 C Elementary charge e h/2T=1.05 x 10-34 Js Planck's constant 3.00 x 108 m s-1 Speed of light...
This problem is concerned with evaluating some improper integrals. In particular you will use an improper integral over an interval of infinite length to evaluate an integral of a function not defined at one end point. This will involve a special function「which arises in many applications in the sciences. a. Evaluate Jo (log z) dr. b. Explain how you would evaluate Jr*(log x)7 dr, but do not actually compute it. Would your method work if the exponent'8, were replaced by...
Use Definition 7.1.1,DEFINITION 7.1.1 Laplace TransformLet f be a function defined for t ≥ 0. Then the integralℒ{f(t)} = ∞e−stf(t) dt0is said to be the Laplace transform of f, provided that the integral converges.to find ℒ{f(t)}. (Write your answer as a function of s.)f(t) = cos(t),0 ≤ t 0)