Gaussian function solutions

The bell-shaped Gaussian function, m) = sap H (\"'3'\")2], is one of the most widely used functions in science and technology. The parameters or and s 1) 0 are prescribed real numbers. Make a program for evaluating this function when m = D, a = 2, and a: = 1. Verify the program's rllult by comparing with hand calculations on a calculator. Show that the inverse Fourier transform of a Gaussian function in k-space F(k) = V2mode *"is another Gaussian function in position space f(x) = Ae *(20") (Hint: the integration technique you need to use here is similar to that given in class for the Fourier transform from /(x) to F(k).)6. The (normalized) Gaussian function is defined as -2 2 g(x) = e 202 where J_ g(x)dx = 1. Note that this implies that the "DC component" of its Fourier transform G(0) = 1. Prove that the Fourier transform is also a Gaussian function of the form G ( w) = e 2 Hint: Use the differentiation with respect to t, and multiplication properties of the Fourier transform, and then integrate both sides