I expected the following formula to hold:
$\int^{2n\pi}_0\cos(\sin t+t/n)dt=0$, for ${}^\forall n\in\mathbb{N},\ n\geq2$
But I can't prove it. Could you please tell me.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityI expected the following formula to hold:
$\int^{2n\pi}_0\cos(\sin t+t/n)dt=0$, for ${}^\forall n\in\mathbb{N},\ n\geq2$
But I can't prove it. Could you please tell me.
You can rewrite the integral as $$ \int_0^{2\pi} \left(\sum_{j=0}^{n-1}\cos\Big(\sin t+\tfrac tn+2\pi \tfrac jn\Big)\right)\,dt. $$ But $\sum_{j=0}^{n-1}\cos\big(a+2\pi \tfrac jn\big)=0$ for all $a$.
In particular, the equality holds if $\sin t$ is replaced by any $2\pi$-periodic function.