Laurent Series for Trig functions

Question

Around a real value, Taylor expansion works but i have no idea how the expansion around infinity is done. Can anyone help show me how the expansion for tangent is done, preferably a derivation or resource that I can learn from. Thanks.

Answer 1

For finding series at $z=\infty$, make the change of variable $w= 1/z$ and find the Laurent Series about $w=0$.

Edit

For your example problem, we're looking at finding the series for $f(z) = z \tan (\pi / z)$ centered at $z=\infty$, and then evaluating at $z=2+4i$ (I assume). Taking the change of coordinates $w= 1/z$ yields the function $g(w) = \frac{1}{w} \tan (\pi w)$ which has a series at $w=0$ of

$$g(w) = \pi +\frac{\pi ^3 w^2}{3}+\frac{2 \pi ^5 w^4}{15}+\frac{17 \pi ^7 w^6}{315}+\cdots$$

Note that $g(w)$ has a removable singularity at $w=0$, so we actually end up with a one-sided Laurent Series (no negative exponents).

Since we want to evaluate at $z=2+4i$, we have that $w= 1/(2+4i) = (1-2i)/10$. Evaluating this series at $w$ gets you home.

Alternatively, we could change our series back into the $z$ variable by $z=1/w$ and evaluate at $z=2+4i$.

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EDIT 2

Per comments below, $i$ is supposed to be a variable and not $\sqrt{-1}$. I'm going to replace $i$ with $z$ because it's headache inducing otherwise. As such, the original function is $$f(z)= (4+2z) \tan \left( \frac{\pi}{4+2 z} \right).$$ As before, we make the change of variables $w= 1/ z$ to get $$g(w) = (4+2/w) \tan \left( \frac{\pi}{4+2/w} \right)$$ which has a series at $w=0$ of
$$g(w) = \pi +\frac{\pi ^3 w^2}{12}-\frac{\pi ^3 w^3}{3}+\left(\pi ^3+\frac{\pi ^5}{120}\right) w^4+ \cdots$$ which is computed with the usual Taylor Series formula. We get back to a series for $f(z)$ centered at $z=\infty$ by using the change of variables $z=1/w$ to get $$f(z) = \pi +\frac{1}{12} \pi ^3 \left(\frac{1}{z}\right)^2-\frac{1}{3} \pi ^3 \left(\frac{1}{z}\right)^3+\left(\pi ^3+\frac{\pi ^5}{120}\right) \left(\frac{1}{z}\right)^4+\cdots$$

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Edit 3

http://www.maths.usyd.edu.au/u/olver/teaching/NCA/09.pdf appears to have a good overview of the theory, starting with analyticity at infinity and moving through Laurent Series over different annuli.