1. Fourier Analysis Fourier Series: A Fourier series is a representation of a function using a series of sinusoidal functions of different “frequencies”. (Recall: Taylor & other power series expansions in Calculus II) They are extremely useful to be used to represent functions of phenomena that are periodic in nature. Fourier Integral & Transform: Similar to Fourier series but extend its application to both periodic and non-periodic functions and phenomena. First developed by the French mathematician Joseph Fourier (1768-1830)

2. Fourier Series Why using Fourier analysis? Analyze the phenomena from time- basis to a frequency-basis analysis. It is simpler to describe a periodic function using Fourier description. Ex: a periodic time series can always be described by using its frequency and amplitude. A periodic function f(x) has a period of p is f(x)=f(x+p), that is, the function repeats itself every interval of length p Trigonometric series: 1, cos(x), sin(x), cos(2x), sin(2x), … These functions all have the period of 2 

4. Euler Formulas

6. Periodic, vanishes All sin(nx)cos(mx) terms vanishAll cos(nx)cos(mx) terms Vanish except when n=m