Charged Particle Optics Theory

Appendix AThe Fourier transform

As a mathematical method, the Fourier transform provides a powerful, simplifying tool for a variety of physical problems. This derives from the fact that a Fourier transform of a function represents the spectral density of the function in the frequency domain. It is a special case in the general theory of Hilbert spaces. Rather than attempt a complete description of this theory, we will confine our attention here only to those aspects that are directly applicable to the present study.

We consider an arbitrary complex function f(x), defined over the range −∞ < x < +∞. We define the Fourier transform f˜(k) as

f˜(k)=∫−∞∞dxe−ikxf(x),⁢(A.1)

where k is called the transform variable, and in general −∞ < k < +∞. We assume ...