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Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] under the condition that the wave function is even? - It is indeed about the number of zeros and not determining the zeros themselves.

Does anybody knows about problems rising in other disciplines like physiscs where the number of such zeros are studied and essential?

Alternatively, if there might be a proof that such method can not generally exist (for instance because of a type of uncertainty principle), it would be also very helpful.

Many thanks

al-Hwarizmi
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    http://www.encyclopediaofmath.org/index.php/Rouch%C3%A9_theorem – Nikita Evseev Mar 27 '13 at 06:08
  • see: partly dealt in another post – al-Hwarizmi May 25 '13 at 09:26
  • Perhaps you would find Riemann-Roch useful? http://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem – Neal Nov 14 '14 at 17:07
  • why/how Rieman-Roch? may you explain more details please? – al-Hwarizmi Nov 14 '14 at 17:48
  • Are you interested in the purely numerical approaches? It would be quite easy to use some numerical rootfinding techniques and then count the roots. This can be done very fast in practice (see here http://www.chebfun.org/docs/guide/guide03.html). – rajb245 Nov 21 '14 at 14:58
  • Put another way, I could write a Matlab function that calculates exactly what you want; internally, it would find all the roots and return the count. As far as you are concerned however, it is the counting function you are looking for. Can you provide some detail about if/why this is not acceptable? – rajb245 Nov 21 '14 at 15:10
  • A numerical approach is actually not what I am looking for (rootfinding algorithms and then counting the loops...). I aim at finding an analytical way (even if a stochastic estimate) how to find the "number" of zeros, without knowing the zeros themselves. I can not find any literature even addressing this in some way or for even some special case. – al-Hwarizmi Nov 21 '14 at 16:20
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    I will just say that this is equivalent to asking: "without calculating any eigenvalues, how many distinct eigenvalues does this matrix have?" The question here shows this: http://math.stackexchange.com/questions/370996/roots-of-a-finite-fourier-series Without more about the nature of the series coefficients, I don't think it is possible to to count eigenvalues/roots without calculating them first. – rajb245 Nov 21 '14 at 19:08
  • this is an interesting comment indeed. My question would be perfectly answered if you would help me to a mathematical proof for what you say. In other Words: when is it possible and when is it not possible to have a count without knowing the zeros/evs? – al-Hwarizmi Nov 21 '14 at 19:11

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