# Give the order of each zero...

#### jzellt

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?

#### chisigma

MHF Hall of Honor
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

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

#### jzellt

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

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#### HallsofIvy

MHF Helper
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.