The latter says that in every neighborhood of an essential singularity " a ", the function " f " takes on " every " complex value, except possibly one, infinitely many times.
22.
:The function e ^ {-z } only has a pole at infinity, and that is an essential singularity, who is saying it has a simple pole ? talk ) 12 : 03, 7 January 2014 ( UTC)
23.
A related definition is that if there is a point a for which f ( z ) ( z-a ) ^ n is not differentiable for any integer n > 0, then a is an essential singularity of f ( z ).
24.
The category " essential singularity " is a " left-over " or default group of singularities that are especially unmanageable : by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner & ndash; poles.
25.
Another way to characterize an essential singularity is that the Laurent series of " f " at the point " a " has infinitely many negative degree terms ( i . e ., the principal part of the Laurent series is an infinite sum ).
26.
If the inner radius of convergence of the Laurent series for " f " is 0, then " f " has an essential singularity at " c " if and only if the principal part is an infinite sum, and has a pole otherwise.
27.
As a matter of interest, I see the wikipedia article on infinitely differentiable non-analytic function'gives that the analytic extension to \ mathbb { C } has an essential singularity at the origin for z \ mapsto e ^ { \ frac {-1 } { z } }-is the same true with z ^ 2 in lieu of z?
28.
I think the standard way to prove that function has an essential singularity is to show that on an arbitrarily small interval around 0, it takes all values between-1 and 1, meaning there is no way to continuously extend the function to 0 ( so it isn't a removable singularity, it's clearly not a pole because sine is bounded on the reals ).
29.
If the principal part of " f " is a finite sum, then " f " has a pole at " c " of order equal to ( negative ) the degree of the highest term; on the other hand, if " f " has an essential singularity at " c ", the principal part is an infinite sum ( meaning it has infinitely many non-zero terms ).
30.
In this case, one or both of the limits \ scriptstyle L ^ {-} and \ scriptstyle L ^ { + } does not exist or is infinite so " x " 0 is an " essential discontinuity ", " infinite discontinuity ", or " discontinuity of the second kind " . ( This is distinct from the term " essential singularity " which is often used when studying functions of complex variables .)
