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.
Apr 20th 2010, 10:12 AM
The complex variable function can be expressed as 'Weierstrass product' in the following way...
That means that is an entire function, i,e. it is analytic on the whole complex plane. From (1) we obtain...
... so that is also an entire function...
Apr 20th 2010, 10:55 AM
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 . The power series for is just that divided, term by term, by z: .