I am very confused on how to obtain the principal part, and in general the Laurent series of functions of complex variable.
I will write an exercise and try to point out where are my doubts.
Consider the function $\tan(z)$ in the annulus $\lbrace3<|z|<4\rbrace$. Let $f(z)=f_0(z)+f_1(z)$ be the Laurent decomposition of $f(z)$, so that $f_0(z)$ is analytic for $|z|<4$ and $f_1(z)$ is analytic for $|z|>3$ and vanishes at $\infty$. (a) Obtain an explicit expression for $f_1$.
Since $\cos(z)=0$ only when $z=\pm(2n+1)\pi/2$ ($n=0,1,2,\dots$), there are no poles inside the annulus. Therefore $f_1(z)=0$. Is this correct?
(b) Write down the series expansion for $f_1(z)$ and determine the largest domain on which it converges.
If I am wrong before, what is the answer to this?
Best Answer
Hints: No, $f_1$ is not zero. If it were, then $\tan(z) = f_0(z)$ would be analytic for $|z| < 4.$
$f_1(z)$ needs to account for the poles within $|z| \le 3.$ There are two: one at $\frac{\pi}{2}$ and the other at $-\frac{\pi}{2},$ and both are simple. So your expression should be something like $$f_1(z) = \frac{C_1}{z - \pi/2} + \frac{C_2}{z + \pi/2}$$ for some constants $C_1,C_2.$