
Complex Power Series
Hi, how are you supposed to represent f(z) = sin(z)/z as a power series of the form sigma c_n z^n ? I have in my notes a formula for this when f is holomorphic in a ball centred at 0, but clearly here f has a singularity at 0 so I don't know what to do.
Thanks

The complex variable function $\displaystyle \sin z$ can be expressed as 'Weierstrass product' in the following way...
$\displaystyle \sin z = z\cdot \prod_{n=1}^{\infty} (1\frac{z^{2}}{n^{2} \pi^{2}})$ (1)
That means that $\displaystyle \sin z$ is an entire function, i,e. it is analytic on the whole complex plane. From (1) we obtain...
$\displaystyle \frac{\sin z}{z} = \prod_{n=1}^{\infty} (1\frac{z^{2}}{n^{2} \pi^{2}})$ (2)
... so that $\displaystyle \frac{\sin z}{z}$ is also an entire function...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$

chisigma, Magus01 wants an infinite sum, not an infinite product! Also, by the way, the word in English is "analytic".
Magus01, the standard power series for sin(z) is $\displaystyle \sum_{n=0}^\infty \frac{(1)^n}{(2n+1)!} x^{2n+1}$. The power series for $\displaystyle \frac{sin(z)}{z}$ is just that divided, term by term, by z: $\displaystyle \sum_{n=0}^\infty \frac{(1)^n}{(2n+1)!}x^{2n}$.