In this problem, you will directly compute the convolution of two Gaussian functions in two steps.

a. Use completing the square to evaluate

\[\int_{-\infty}^{\infty} e^{-\alpha t^{2}+\beta t} d t\]

b. Use the result from part a to directly compute the convolution in Example 8. 16:

\[(f * g)(x)=e^{-b x^{2}} \int_{-\infty}^{\infty} e^{-(a+b) t^{2}+2 b x t} d t\]

Data from Example 8.16

Convolution of two Gaussian functions \(f(x)=e^{-a x^{2}}\).