Sine Curve

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Sine Curve

Drag the slider to change the frequency:

Lissajous figures are parametric curves where both x(t) and y(t) are sine functions. If the ratio of the frequencies is rational, the curve will always eventually close. If it is irrational, the curve will never close and eventually fill a region.

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A Lambert series has the form Sigma (k=1 to inf) [ a_{k}*z^{k} / (1-z^{k})] . For bounded coefficients
a_{k}
, the series converges in the unit disk and most poles lie on the unit circle. These series play an important role in analytic number theory and are related to the divisor function, Jacobi theta functions, the Möbius function, Euler's totient function, the Liouville function, the Ramanujan theta function, and the Riemann zeta function.

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References

[1] E. Wegert, Visual Complex Functions: An Introduction with Phase Portraits, New York: Birkhäuser, 2012.

[2] Mr. Wizard. "How can I generate this "domain coloring" plot?" Mathematica StackExchange. (Feb 5, 2015) mathematica.stackexchange.com/questions/7275/how-can-i-generate-this-domain-coloring-plot.

[3] J. Arndt, "On Computing the Generalized Lambert Series." arxiv.org/abs/1202.6525.