Give the order of each zero...

Feb 2008
535
4
Give the order of each of the zeros of the function:

sinz / z

Can someone please expalin or show the steps needed to do this?

Thanks in advance...
 

chisigma

MHF Hall of Honor
Mar 2009
2,162
994
near Piacenza (Italy)
Considering the expansion of the function as 'infinite product'...

\(\displaystyle \frac{\sin z}{z} = (1-\frac{z}{\pi}) (1+\frac{z}{\pi}) (1-\frac{z}{2 \pi}) (1+\frac{z}{2\pi}) \dots\) (1)

... You can easily verity that the zeroes are at \(\displaystyle z = k \pi\) with \(\displaystyle k \ne 0\) and each of them has order 1...

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)
 
Feb 2008
535
4
Thanks for the quick reply.

I guess I'm really behind here but how do you find that

sinz/z = (1 - z/pi)(1 + z/pi)(1 - z/2pi)... ?

Also, what tells you that the zeros are of order 1,2,3,...?

Thanks
 
Last edited:

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
His point was that if you write it as such an infinite product, you can see that each zero gives one factor and so has order one.