I have a doubt with the singularities and poles of a function, for example in the next complex function:
with a constant. The function is zero when
For this value of x, is it a singularity or a pole? If it is a pole, of which order? Is it possible to calculate the residue for ?
I quote from Churchill et al Complex Variables and Applications:
"... a point is called a singular point of a function f if f faails to be analytic at but is analytic at some point in every neighbourhood of . A singular point is said to be isolated if, in addition, there is some neighbourhood of throughout which f is analytic except at the point itself."
A branch point is a singular point - what it's not is an isolated singular point.
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There are two categories of singularity:
1. Isolated singularity.
The point is an isolated singularity of if is not analytic at but is analytic in some deleted neighbourhood of .
There are three types:
i. Removable. exists and is finite. Alternatively, the Laurent series expansion of about has no principle part.
Eg. has a removable singularity at z = 0.
ii. Pole. is analytic at for some positive integer m. The smallest such m is called the order of the pole. Alternatively, the principle part of the Laurent series expansion of about terminates after m terms.
Eg. has a simple pole at z = 1.
iii. Essential. It's neither removable nor a pole. Alternatively, the principle part of the Laurent series expansion of about does not terminate. Alternatively, does not exist (NB. Picard's Theorem).
Eg. has an essential singularity at z = 0.
2. Non-isolated singularity.
If the singular point is not isolated, then it is a non-isolated singularity.
Eg. The branch point z = 0 of is a non-isolated singularity. In fact, any branch point of a multiple-valued function is a non-isolated singularity.
Yes. But it should be noted that branch cuts are defined by convention ..... You say , I could just as well define the branch cut to be in which case the points in are not singular .....
The branch point is (the only point that is) common to all branch cuts .... This makes it fundamentally different from all other points on a branch cut.