Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 6A: On Restricted Fourier Series Expansion

A periodic function expressed by

(6A.1)$$f\left(t\right)=f\left(t+T\right),$$

where T is the period, may be expressed in Fourier series as

(6A.2)$$f\left(t\right)=a_0+{\displaystyle \sum }_{n=1}^{\infty }{a}_{n}cos\left(n{\mathrm{\omega }}_{0}t\right)+{\displaystyle \sum }_{n=1}^{\infty }{b}_{n}sin\left(n{\mathrm{\omega }}_{0}t\right).$$

In Equation 6A.2,

(6A.3)$${\mathrm{\omega }}_0=\frac{2\mathrm{\pi }}{T},$$

(6A.4)$$a_0=\frac{1}{T}\underset{-T/2}{\overset{T/2}{{\displaystyle \int }}}f\left(t\right)\text{d}t,$$

(6A.5)$$a_n=\frac{2}{T}\underset{-T/2}{\overset{T/2}{{\displaystyle \int }}}f\left(t\right)cos\left(n{\mathrm{\omega }}_{0}t\right)\text{d}t,$$

(6A.6)$$b_n=\frac{2}{T}\underset{-T/2}{\overset{T/2}{{\displaystyle \int }}}f\left(t\right)sin\left(n{\mathrm{\omega }}_{0}t\right)\text{d}t.$$

In trying to solve the Laplace equation in rectangular coordinates with specified boundary conditions, we need Fourier series expansion of an arbitrary function defined over an interval. The boundary conditions may require use of sine terms-only, odd harmonics-only, and so on. The following example [1] illustrates how the function will look in its entire period and what the period is.

Example 6A.1

A function x(t