Every function with an infinite Taylor series results in a series with an infinite negative part of the Laurent series by substituting ##z\mapsto 1/z.## Sine, cosine, logarithm, etc.LagrangeEuler said:Yes, I know that. But I do not know how to find those examples.
Excellent. We have to add that the original Taylor series must have an infinite radius of convergence as your examples do.fresh_42 said:Every function with an infinite Taylor series results in a series with an infinite negative part of the Laurent series by substituting ##z\mapsto 1/z.## Sine, cosine, logarithm, etc.
An essential singularity is a type of singular point of a complex function where the behavior of the function is particularly chaotic. More formally, a point \( z_0 \) is classified as an essential singularity if the limit of the function does not exist as \( z \) approaches \( z_0 \), and the function does not approach any finite value or infinity. This is in contrast to removable singularities and poles, where the function behaves more predictably.
To identify an essential singularity, you can use the behavior of the function's Laurent series expansion around the point in question. If the series has infinitely many negative powers of \( (z - z_0) \) and does not converge to a finite limit as \( z \) approaches \( z_0 \), then \( z_0 \) is an essential singularity. Additionally, the Casorati-Weierstrass theorem states that in any neighborhood of an essential singularity, the function takes on every complex value, with possibly one exception.
The Weierstrass-Casorati theorem highlights the unpredictable nature of functions near essential singularities. It states that if \( f(z) \) has an essential singularity at \( z_0 \), then in every neighborhood of \( z_0 \), the function can take on every complex value, except possibly one. This theorem illustrates the extreme variability of functions near essential singularities and emphasizes why they are considered critical points in complex analysis.
A classic example of a function with an essential singularity is \( f(z) = e^{1/z} \) at \( z = 0 \). As \( z \) approaches 0, the function exhibits wildly oscillating behavior, taking on every complex value infinitely often, which demonstrates the characteristics of an essential singularity. The function does not approach a finite limit or infinity, further confirming the nature of the singularity at that point.
The key difference between an essential singularity and a pole lies in the behavior of the function near these points. At a pole, the function approaches infinity as \( z \) approaches the singularity, and the Laurent series around the pole has a finite number of negative powers. In contrast, at an essential singularity, the function does not approach any specific value (finite or infinite) and has infinitely many negative powers in its Laurent series, indicating a more erratic behavior.