Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have no practical reason for wanting to do this, but I was wondering why the Fourier series for $\sin x$ is the identical zero function.

I'm probably doing something wrong or missing some important condition.

Could someone help me see?

share|cite|improve this question
What have you done? – draks ... Oct 16 '12 at 10:48
Why do you believe it is the zero function? $\sin(x)=1\cdot \sin(x)+0\cdot \sin(2x)+0\cdot \sin(3x)+\cdots$. – Per Manne Oct 16 '12 at 10:48
@draks Oh then I must've gotten the coefficients wrong. The period of $\sin x$ is $2\pi$, so I got*+integrate+sin+x+sin+nx+from+0+to+p‌​i as the general term for the coefficients, which I figure is zero since $\sin (n\pi)$ is zero for all integers n. – Ryan Oct 16 '12 at 10:52
But for $n=1$ you need to integrate $\sin^2 x$. – Javier Oct 16 '12 at 10:55
up vote 11 down vote accepted

Wolfram gives the following:

$$ \frac2{\pi}\int_{0}^{\pi} \sin x \sin (nx)\ dx = -\frac{2\sin(n\pi)}{\pi(n^2-1)} $$

You are almost correct in that this is zero for all $n$ because $\sin(n\pi) = 0$ for every integer. But when $n=1$, the formula doesn't work, because the $n^2-1$ in the denominator becomes zero too. You need to consider that as a special case:

$$ \frac2{\pi}\int_{0}^{\pi} \sin x \sin (1x)\ dx = \frac2{\pi}\int_0^{\pi} \sin^2 x \ dx = \frac2{\pi} \frac{\pi}{2} = 1. $$

share|cite|improve this answer
Oh yes, I just saw that too. The special case needed to be introduced at the point of trying to integrate $\cos( (1-n)x)$. Heehee! Thanks Javier. – Ryan Oct 16 '12 at 11:17
We can still use the formula. Take the limit $n\to 1$. (+1) by the way. – user26872 Oct 16 '12 at 11:18

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.