- #1
LagrangeEuler
- 717
- 20
Can you give me two more examples for essential singularity except [tex]f(z)=e^{\frac{1}{z}}[/tex]? And also a book where I can find those examples?
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 singularity in complex analysis where a function behaves in a very irregular manner near a point in the complex plane. At an essential singularity, the function cannot be defined by a Laurent series with a finite number of terms.
Essential singularities can be identified by observing the behavior of a function near a point in the complex plane. If the function has an essential singularity at a point, it will have an infinite number of terms in its Laurent series expansion around that point.
Some common examples of functions with essential singularities include the exponential function e^1/z and the sine function sin(1/z). These functions exhibit complex behavior near the origin in the complex plane due to their essential singularities.
Essential singularities differ from poles in that poles are characterized by having a finite number of terms in their Laurent series expansions, while essential singularities have an infinite number of terms. Poles also have a well-defined residue, whereas essential singularities do not.
Essential singularities play a crucial role in understanding the behavior of complex functions, particularly near points where the function exhibits irregular behavior. They provide insight into the structure of functions and their properties in the complex plane, helping to analyze and classify different types of singularities.