Proving Simple Pole of $\frac{1}{1-2^{1-z}}$ at $z=1$

[mathwonk]

sorry i waS CRANKY TODAY.

but it is kind of odd to know what a simple pole is and not know what a simple zero is.

a meromorphic function has a simple pole at a if it looks locally near a, like

1/(z-a) times a holomorphic function which is not zero at a.

it has a simple zero at a if it looks locally at a like (z-a) times a holomorphic function which is not zero at a.hence obviously f has a simple pole iff 1/f has a simple zero.

as you realized, since a holomorphic function has a taylor series, whose coefficient of (z-a) is its first derivative at a, a holomorphic f has a simple zero at a iff f(a) = 0 and f'(a) is not zero.so to show the function above has a simple pole at z=1, it is much easier to turn it upside down and show the reciprocal has a simple zero, which can be checked by taking a derivative.but somebody is teaching you amiss, if they have you doing meromorphic functions and poles and have not even taught you about using derivatives to compute the order of a zero, which is easier and more fundamental.

a holomorphic function, e.g. a polynomial, has a zero of order at least k at a, iff if is divisible by (z-a)^k, (with holomorphic quotient),

iff its first k derivatives (0'th through k-1'st) are zero at a.

a simple zero is a zero of order one. i would think this would be familiar even from high school algebra.

 

[Zurtex]

Thanks a lot mathwonk, I know the author of the thread in real life, I don't think either of us have come across simple zero's before but I think we both understood what you were on about straight away, very useful thanks

 

[mathwonk]

you are quite welcome.

there is also a geometric version of the order of a zero or pole.

a point a is a zero of f of order k iff the inverse image of a small punctured disc D centered at 0, intersects a small nbhd of a, in a set that maps exactly k to one onto D.

since reciprocation is an isomorpism from a nbhd of 0 to a nbhd of infinity, a pole of order k at a, means some punctured nbhd of a maps exactly k to one, onto the exterior of a large disc, i.e. ointo a small opunctured disc about infinbity.