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# What is Laurent series used for?

## What is Laurent series used for?

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

What is the difference between Taylor and Laurent series?

Summary. A power series with non-negative power terms is called a Taylor series. In complex variable theory, it is common to work with power series with both positive and negative power terms. This type of power series is called a Laurent series.

How do you find the coefficient in Laurent series?

c−1=12πi∮γf(t)dt.

### How do you find the region of convergence in Laurent series?

Its radius of convergence around z=−2, and therefore the radius of convergence of your Laurent series, is simply the distance from −2 to the nearest non-differentiable point, i.e. z=−1.

What is the principal part of a Laurent series?

with the series convergent in the interior of the annular region between the two circles. The portion of the series with negative powers of z – z 0 is called the principal part of the expansion.

What is Lawrence Theorem?

For her favorite theorem, Dr. Lawrence chose the classification of compact surfaces, one of the best theorems from a first topology class. The classification theorem states that all surfaces that satisfy some mild requirements are topologically equivalent to a sphere, a sum of tori, or a sum of projective planes.

#### What is the principal part of Laurent series?

The portion of the series with negative powers of z – z 0 is called the principal part of the expansion. It is important to realize that if a function has several ingularities at different distances from the expansion point , there will be several annular regions, each with its own Laurent expansion about .

What is principal part of Laurent series?

What is meant by removable singularity?

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.