[SOLVED] Laurent series

Bop

Hello, I want to calculate laurent series and its convergence radius of:

$$\displaystyle f(z)=\frac{1}{sin({\pi z})}$$

Here it is what i've done, I'm not sure if we can do it in this way:

$$\displaystyle sin z =\sum \frac{(-1)^{n}}{(2n+1)!}z^{2n+1}=z-\frac{z^3}{3!}+\frac{z^5}{5!}-O(z^7)$$ for $$\displaystyle -1<z<1$$

so:

$$\displaystyle sin (\pi z) =\sum \frac{(-1)^{n}}{(2n+1)!}(\pi z)^{2n+1}=\pi z-\frac{(\pi z)^3}{3!}+\frac{(\pi z)^5}{5!}-O((\pi z)^7)$$ for $$\displaystyle -1<z<1$$

so:

$$\displaystyle \frac{1}{sin (\pi z)} =\sum \frac{(2n+1)!}{(-1)^{n}}\frac{1}{(\pi z)^{2n+1}}=\frac{1}{\pi z}-\frac{3!}{(\pi z)^3}+\frac{5!}{(\pi z)^5}-\frac{1}{O((\pi z)^7)}$$ for $$\displaystyle -1<z<1$$

Is it right?

Thank you.

stokastik

warning
$$\displaystyle \sum \frac{A}{B} \neq (\sum \frac{B}{A})^(-1)$$

chisigma

MHF Hall of Honor
We can start from the series expansion...

$$\displaystyle \frac{\sin (\pi z)}{\pi z} = 1 - \frac{(\pi z)^{2}}{3!} + \frac{(\pi z)^{4}}{5!} - \dots$$ (1)

... and then, setting...

$$\displaystyle f(z) = \frac{\pi z} {\sin (\pi z)} = a_{0} + a_{1} z + a_{2} z^{2} + \dots$$ (2)

... find the $$\displaystyle a_{n}$$ imposing...

$$\displaystyle f(z) \frac{\sin (\pi z)}{\pi z} = (a_{0} + a_{1} z + a_{2} z^{2} + \dots) \{ 1 - \frac{(\pi z)^{2}}{3!} + \frac{(\pi z)^{4}}{5!} - \dots\} = 1$$ (3)

From (3) now we derive directly...

$$\displaystyle a_{0} = 1$$

$$\displaystyle a_{1} =0$$

$$\displaystyle a_{2} - a_{0} \frac{\pi ^{2}}{3!} = 0 \rightarrow a_{2} = \frac{\pi^{2}}{6}$$

$$\displaystyle a_{3} =0$$

$$\displaystyle a_{4} - a_{2} \frac{\pi^{2}}{3!} + a_{0} \frac{\pi^{4}}{5!}=0 \rightarrow a_{4} = \frac{7 \pi^{4}}{360}$$ (4)

... and so one, so that is...

$$\displaystyle \frac{\pi z}{\sin (\pi z)} = 1 + \frac {\pi^{2} z^{2}} {6} + \frac{7 \pi ^{4} z^{4}}{360} + \dots$$ (5)

... and now, deviding (5) by $$\displaystyle \pi z$$, we obtain...

$$\displaystyle \frac{1}{\sin (\pi z)} = \frac{1}{\pi z} + \frac {\pi z} {6} + \frac{7 \pi ^{3} z^{3}}{360} + \dots$$ (6)

The Laurent series (6) converges for $$\displaystyle 0 < |z| < 1$$ ...

Kind regards

$$\displaystyle \chi$$ $$\displaystyle \sigma$$

Bop