# Complex Power Series

• Apr 20th 2010, 06:31 AM
Magus01
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
• Apr 20th 2010, 09:12 AM
chisigma
The complex variable function $\sin z$ can be expressed as 'Weierstrass product' in the following way...

$\sin z = z\cdot \prod_{n=1}^{\infty} (1-\frac{z^{2}}{n^{2} \pi^{2}})$ (1)

That means that $\sin z$ is an entire function, i,e. it is analytic on the whole complex plane. From (1) we obtain...

$\frac{\sin z}{z} = \prod_{n=1}^{\infty} (1-\frac{z^{2}}{n^{2} \pi^{2}})$ (2)

... so that $\frac{\sin z}{z}$ is also an entire function...

Kind regards

$\chi$ $\sigma$
• Apr 20th 2010, 09:55 AM
HallsofIvy
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 $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1}$. The power series for $\frac{sin(z)}{z}$ is just that divided, term by term, by z: $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n}$.