- #1
kent davidge
- 933
- 56
I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero.
On one hand, as we can in principle choose whatever values we like for ##m## and ##n## (as long as they are integer numbers), if we let ##m \rightarrow -\infty## and ##n \rightarrow +\infty##, then the integral ##\int_{-\infty}^{+\infty} e^{ixa}dx## should vanish.
On the other hand, this is absurd, since I know these exponentials even work as an orthonormal basis in Fourier expansions.
I presume my mistake is in taking those limits. What you think?
On one hand, as we can in principle choose whatever values we like for ##m## and ##n## (as long as they are integer numbers), if we let ##m \rightarrow -\infty## and ##n \rightarrow +\infty##, then the integral ##\int_{-\infty}^{+\infty} e^{ixa}dx## should vanish.
On the other hand, this is absurd, since I know these exponentials even work as an orthonormal basis in Fourier expansions.
I presume my mistake is in taking those limits. What you think?