The Fourier series of an odd function only has sine terms—all the cosine coefficients are zero—and so the Fourier series is a sine series.

What is the sine series for a sine function? If the frequency is an integer, then the sine series is just the function itself. For example, the sine series for sin(5*x*) is just sin(5*x*). But what if the frequency is not an integer?

For an odd function *f* on [-π, π] we have

where the coefficients are given by

So if λ is not an integer, the sine series coefficients for sin(λ*x*) are given by

The series converges slowly since the coefficients are *O*(1/*n*).

For example, here are the first 15 coefficients for the sine series for sin(1.6*x*).

And here is the corresponding plot for sin(2.9*x*).

As you might expect, the coefficient of sin(3*x*) is nearly 1, because 2.9 is nearly 3. What you might not expect is that the remaining coefficients are fairly large.

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