Principles of Quantum Mechanics

Appendix B

Some Supplementary Topics

B.1  FOURIER TRANSFORM

B.1.1  Fourier Series

Let us consider a periodic single valued function f(x) in the interval –π ≤ x ≤ π [such that f(x + 2π) = f(x)]. The Fourier series corresponding to f(x) is defined as

This series converges in the said interval (–π, π), if f(x) and f′(x) [= (∂f /∂x)] are continuous. Multiplying Eq. (B.1) by cos(mx) and integrating over –π to π, we get

Again multiplying (B.1) by sin (mx) and integrating over −π to π, we get

The series (B.1) may also be written in an alternative form ...