Gaussian function variable

Example 2-4 Gaussian function s(t) = (2.27) which is the standard Gaussian function with the mean time t, The Fourier trans- form is s(w) - Jamexp -"(t-1 ) expi-jothat = exp. -zoo +jot. (2.28) The formula (2.28) usually is remembered as the Gaussian characteristic function, Eq.(2.28) shows that the Fourier transform of the Gaussian function is also Gauss- ian. The time-shift in (2.27) corresponds to the phase-shift in (2.29). Moreover, the variance of the frequency representation in (2.28) is the reciprocal of the time vari ance in (2.27), The narrower the spread in the time domain is, the wider the spread in the frequency domain, or vice versa. Unlike the sinusoidal or rectangular pulse functions discussed earlier, the Gaussian function is localized in both time and fre- quency. As we shall see later, among all possible functions, the Gaussian function is optimally concentrated in joint time and frequency domain.Problem 1 Consider a Gaussian function in the form: (x) = Ae-012/2 (1) a) Normalize this function, i.e., find the normalization constant A. b) Show that this function is the eigenfunction of a one-dimensional harmonic oscillator. c) Determine the parameter o via the mass and angular frequency of the oscillator. d) Derive the energy corresponding to this eigenfunction."\"2 is widely used in mathematics, statistics, and :r:2 1. The Gaussian function f(:r) = 6 engineering. We will derive the Fourier transform of the Gaussian function f(:r) = e by the following steps. on (a) Evaluate the improper integral I = f 3'12 (is. Hint: First square the integral woo and then use the polar coordinate to evaluate the double integral, i.e, 1'2 = f e\"? dz: 8-\": dy = f / e"\"2+yaldmdy. a: 5 (c) Find the Fourier transform of f (:1?) = 6-12. Hint: Let u: = FIf](w) be the Fourier transform of f (3:) Then take the derivative of re). You may need to use the property that F[f'](w) = in[f](w). (b) Solve the ODE y' = y with the initial condition y(0) = yo