However, it has some irrational properties. According to WolframAlpha, \(\lim_{x\to\infty}z_x^x = e^{2i}\). The function \(f\) is probably injective for some values of \(a\).
Sunday, June 22, 2014
A rational sinusoidal function
It is easy to get confused by the irrational nature of the sin function. One might think that you need irrational numbers to create a sinusoidal graph, but that's false. Consider \(z_a = (a+i)^2/(a^2+1) \quad (a \in \mathrm{Rat})\). It lies on the unit circle in the complex plane and its coordinates are rational. Here is a rational sinusoidal function: \(f(x) = \operatorname{Im}(z_a^x) \quad (x \in \mathrm{Int})\). This is what it may look like:
Subscribe to:
Post Comments (Atom)
To prove WolframAlpha's claim, cancel x^2 in the numerator and denominator of z_x and use the limit definition of e^x.
ReplyDeleteAnother property is (z_10000)^31417 ~= 1 and so on.
ReplyDeleteBut that's closely related to the e^2i limit.
Delete