Define $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$
Where is this function defined and analytic?
My answer:
Our function is analytic wherever it has a convergent power series. Since (I am assuming) $\sin(z)$ and $\cos(z)$ are analytic the quotient is analytic wherever $\cos(z) \not\to 0$???
Is there more detail to this that I am missing? Without Cauchy is there a way to determine analyticity etc…
Further more, once we have found the first few terms as I have
$$z+\frac{z}{3}+\frac{2}{15}z^5$$
How do we make an estimate of the radius of convergence? Do I make an observation that the terms are getting close to $0$ and say perhaps the $nth$ root is heading there as well, $\therefore$ $R=\infty$. I'm fairly lost on this part.
Thanks for your help!
Best Answer
The formulation of your question reveals a bit of confusion concerning the issue of analyticity. A function can be analytic beyond its radius of convergence; of course it is not defined there by its power series by rather by analytic continuation. The tangent is a good example of this. The radius of convergence is the distance to the nearest zero of cosine, namely $\pi/2$, but the function is analytic everywhere except for points where cosine vanishes.