$\cot(x)\,$ in the large $x$ limit?

Question

I couldn't find asymptotic forms of trigonometric functions in any Math Table.

In particular, I am trying to find $\;\cot(a x)\;$ in large $x$ limit.

thanks,

Answer 1

Notw that $cot(x)$, with period $\pi$, is periodic.

To see this, it helps to remember that $\;\cot(ax) = \dfrac{\cos(ax)}{\sin(ax)},\;$ and recall what happens to $\cot x\,$ each time $\,x \to k\pi,\;k \in \mathbb Z:$

$\qquad\qquad\quad \text{Graph of}\;\;{\bf f(x) = \cot x}$

$\qquad\qquad$

and hence, what happens as $\,ax \to \dfrac {k\pi}{a}.\;$ (For $|a| > 1$, the period decreases, for $|a| < 1$, the period increases.)

Note: $\quad\cot(ax)\,$ does not have any associated asymptotic expansion for large values of $\,x$.

Answer 2

for

$$z = x + iy$$ and as $$y \to \infty$$ we have the following series representation $$cot(z) = -i[1+2 \sum_{k = 1}^{\infty} \exp(2 i k z) ]$$ and once you set $$x$$ to 0, this helps you to get an asymptotic form for $cot(z)$ as $z \to \infty$.