# Fourier Series Of Functions With Arbitrary Periods

**Fourier Series of Functions with Arbitrary Periods:
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So far, we’ve worked with 2π-periodic functions for convenience. Similar results exist for functions having any period. These results can be obtained in a similarmanner. However, there is an easier way, one calculus II students are familiar with: substitution. One way to obtain a new series representation is to perform a substitution in a known series. Suppose that f is a function with period** T = 2p > 0 **for which we want to find a Fourier series. In other words **f (x 2p)** **= f (x)**. If we let** t = πx / p** and we define:

Then g has period that is 2π. So, g has a fourier series representation;

Using the substitution t = πx/p, we obtain

So, we have the following;

**Theorem-27 **Suppose that f is a 2p - periodic function piecewise smooth function. The fourier series of ff is given by;

The Fourier series converges to f (x) if f is continuous at x and to otherwise f is either even or odd.