Singularities of a Complex Function

In summary, the region of validity for the given complex function is when z is not equal to 0 and z is not equal to -2. The singularities occur at z = 0 and z = -2, and in order to find the absolute value of z3, you should review the algebra of the roots and powers of a complex number.
  • #1
ThLiOp
9
0

Homework Statement



What are the region of validity of the following?

1/[z2(z3+2)] = 1/z3 - 1/(6z) +4/z10

Homework Equations

The Attempt at a Solution



Knowing that this is the expansion around z=0, I am trying to find the singularities of the complex function.

Which is when z2(z3+2) = 0

I have z = 0, then (z3+2) = 0

(z3+2) = 0

z3 = -2 = r3ei3θ

In order to find r, would I take the absolute value of z3?

r = |z3| = |-2| = sqrt[(-2)2] = 2?
 
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  • #3
ThLiOp said:
r = |z3|
I don't think that's what you meant. Try again.
 

FAQ: Singularities of a Complex Function

What is a singularity of a complex function?

A singularity of a complex function is a point in the complex plane where the function is not defined or where it becomes infinite. It is an essential concept in complex analysis and is often associated with the behavior of a function near that point.

What are the different types of singularities?

The three main types of singularities are removable, essential, and poles. A removable singularity is a point where the function can be extended to be continuous. An essential singularity is a point where the function has infinite oscillation. A pole is a point where the function becomes infinite.

How are singularities classified?

Singularities are classified based on their behavior near the point. This includes the order of the singularity, which is determined by the highest power of the variable in the Laurent series expansion, and the type of singularity, which is determined by the convergence or divergence of the Laurent series.

What is the significance of singularities in complex analysis?

Singularities play a crucial role in complex analysis as they can reveal important information about the behavior of a function. They are also used to classify and analyze the properties of complex functions, such as analyticity and differentiability.

How are singularities handled in complex function analysis?

In complex function analysis, singularities are often handled through techniques such as the Cauchy integral theorem and residue theorem. These tools allow us to evaluate complex integrals and understand the behavior of a function near a singularity.

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