Deta function Gaussian solutions

(1 point) The following function is called a Gaussian Function. Gaussian functions are used In statistics, Image processing, signal processing, and in mathematics to solve heat and diffusion equations. f(s) - de mi Find the first and second derivative of the given Gaussian function when a = 3, b = 1, and c == 3 f'(=) = -(x-1) (12er(-((x-1)*2)/(9))(3)) f" (x) = -((2e)~(-((x-1)-2)(9))(-2x*2+4x+7))/(27) NOTE: It will be much easier to type your answer if you simplify the derivative functions (but is not necessary). Hint:Show that the Fourier transform of a Gaussian function is also a Gaussian function. Note that a Gaussian function, g(x), is given as follows: g (x) = e-ax2, a 20 And the Fourier transform of a function, f(x), is given by F(f (x)} = F(w) = | f(x)e -jux dx 1-00 (Hint: Start by differentiating the integral formula of the Fourier transform of a Gaussian w.r.t w)Question 2 (Analytical Section) Show that the Fourier transform of a Gaussian function is also a Gaussian function. Note that a Gaussian function, g(x), is given as follows: g(x) = e-ax2 a 20 And the Fourier transform of a function, f (x), is given by Fff (x)} = F(w) = f(x)e-jux dx (Hint: Start by differentiating the integral formula of the Fourier transform of a Gaussian w.r.t w)1 The Dirac o function The delta function can also be described by a Gaussian function 6(x) = lim da (@) = lim exp 0-0 0-0 aV 2TT 2a Use the definition of the o function Eq.(1) to prove the following identities. S(x - xo)f(x) dx = f(x0) - DO Please use Gaussian .OO function to calculate ! 8' (x) f (2) = -f'(2) - Do