The behavior of $f(x,y) = r\cos(n\theta)$ for $n \geq 3$ odd, or $n$ even and $n = 1$.

Question

I wonder whether one can write the function $f(x,y) = r\cos(n\theta)$ in the form of $x$ and $y$ for $n \geq 3$ odd, or $n$ even and $n = 1$. Of course the $r$ and $\theta$ are defined as $x = r\cos(\theta)$ and $y= r\sin(\theta)$.

Background: The original question asks some properties about the function $$ f(x,y) = r\cos(3\theta) = \frac{x^3-3xy^2}{x^2+y^2} $$ with $f(0,0) = (0,0)$. I showed that it is continuous, has every directional derivative at $(0,0)$, and the directional derivatives do not lie in a common plane. The question then asks to generalize the cases for all types of $n$.

FYI, a fairly closely related question to what you're asking, and what KaviRamaMurthy's answer below suggests, is Identities on $\cos n\theta$ and $\sin n\theta$. — John Omielan, May 01 '20 at 23:28

score 0 · Accepted Answer · answered May 01 '20 at 23:25

0

Hint: $f(x,y)=\Re (re^{in \theta})=\frac 1 {r^{n-1}} \Re (x+iy)^{n}$. Expand $(x+iy)^{n}$ by Binomial Theorem.

answered May 01 '20 at 23:25

Kavi Rama Murthy

311,013

The behavior of $f(x,y) = r\cos(n\theta)$ for $n \geq 3$ odd, or $n$ even and $n = 1$.

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