Standard Integrals - 2

Standard Integrals - 2

 This result is quite handy while dealing with the exponential functions, especially while finding its the Fourier co-efficients. This useful result can be easily derived  by completing the square in the exponent and then using the results obtained from the previous post regarding standard integrals. (You can also derive this result, applying product rule & standard integrals)

 

As the limits of integration are infinite, there is essentially no effect for the shifting of the function from x to x-b/2a. (i.e substituting y=x-b\2a and simplifying the integral)

 

Side note: This result is quiet intuitive, if we look into the variation of the function as we change the parameters a and b. For +ve a and b, exp(bx) is an unbounded function & it is the exp(-ax^2) part which brings down the value of the function. Hence, 'a' reduces & 'b' drastically enhances the the total area covered (integral) by the function exp(-ax^2+bx).
 
This result along with Fourier Transform will be used a few times in electromagnetics -to analyze dispersion of (modulated & unmodulated) Gaussian signals and in quantum mechanics to investigate the time evolution of a Gaussian wavepacket & in understanding uncertainty principle.