How do I solve the following equation for $f(\cdot)$?
$f(x)+\frac{1}{n}f(nx)=\sin(x)$
That is, how do I create a function which, when combined with its nth harmonic, will be a sine wave?
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1 Answer
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$$\sum_{k=1}^\infty (\frac{-1}{n})^{k-1}\sin(n^kx)$$
answered May 17, 2019 at 0:26
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$\begingroup$ Can you provide details on how to solve this or point me to somewhere with more information? I'd like to learn more. $\endgroup$– mrplantsCommented May 17, 2019 at 0:33
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2$\begingroup$ @mrplants $$f(x)+\frac{1}{n}f(nx)=\sin(x)\\ \frac{1}{n}f(nx)+\frac{1}{n^2}f(n^2x)=\frac{1}{n}\sin(nx) \\\frac{1}{n^2}f(n^2x)+\frac{1}{n^3}f(n^3x)=\frac{1}{n^2} \sin(x) \\\...$$Now, first equation minus second plus third and so on... $\endgroup$– Nick SCommented May 17, 2019 at 0:38
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$\begingroup$ Does this function have a name or any special properties? Anywhere I can learn more about this? $\endgroup$– mrplantsCommented May 17, 2019 at 0:58
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$\begingroup$ Okay, walked through it on paper and just have a small edit: the answer is actually $\sum_{k=0}^\infty\left(\frac{1}{n}\right)^k\sin(n^kx)$ $\endgroup$– mrplantsCommented May 17, 2019 at 1:22