# A pure sine wave vs. the infinite sum of its harmonics

Is a pure sine wave equivalent to the sum of its every harmonic (up to infinity and without the fundamental lets assume)?

Moreover, if it is so, is this the reason why all the harmonics are present in an open string played or other parts of the instrument played have a role?

• A pure sine wave doesn't have any harmonics, by definition. That's why it's called pure. – knzhou Jul 7 '18 at 16:13
• This is a very confusing question. The "harmonics" are all pure sine (or cosine, or exp(i*something)) wave. No mixing required to produce a sine. I think you are getting terms mixed up. Consider editing this or deleting it. Even though it has answers they can't possible the answer to the question. – ggcg Jul 11 '18 at 3:03

Since $e^{ik\omega t}$ and $e^{im\omega t}$ are orthogonal over any interval of length $2\pi/\omega$ once cannot decompose $e^{ik\omega t}$ into the linear sum of $e^{im\omega t}$ if $k\ne m$ for any $k$, specifically the decomposition $\sum_{m,m\ne 1} A_m e^{im\omega t}$ cannot equal $e^{i \omega t}$.