What is the Fourier transform of (x)? Using Plancherels theorem, show that This formula gives any respectable

What is the Fourier transform of δ(x)? Using Plancherel’s theorem, show that

This formula gives any respectable mathematician apoplexy. Although the integral is clearly infinite when x = 0, it doesn’t converge (to zero or anything else) when x ≠ 0, since the integrand oscillates forever. There are ways to patch it up (for instance, you can integrate from -L to +L, and interpret Equation 2.147 to mean the average value of the finite integral, as L → ∞) . The source of the problem is that the delta function doesn’t meet the requirement (square-integrability) for Plancherel’s theorem (see footnote 42). In spite of this, Equation 2.147 can be extremely useful, if handled with care.

Transcribed Image Text:

8(x) = 1 2π p+∞o Tx 88 eikx dk. (2.147)