6.3. Properties of the Dirac delta-function

Combining the definitions of the Fourier transform and the inverse Fourier transform we can write (suitably arranged)

This is very instructive since the second term, which in the delta-function notation can be written simply as δ (t - t'), has the property of selecting out just one point from the infinite integral and assigning a finite value to the result. This leads to an alternative definition of the delta-function, namely

except for t = t' where the function is infinite or more rigorously