Laurent series and partial fractions

In summary, to find the Laurent series of sin(2z)/(z^3) in [z]>0, you do not need to use partial fractions. Instead, think of a way to write the sine function as a power series, as a Laurent series is unique. The denominator does not need to be rewritten.
  • #1
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Homework Statement



find the laurent series of sin(2z)/(z^3) in [z]>0

Homework Equations





The Attempt at a Solution


I am completely confused. I can understand some of the examples given on laurent series, like using partial fractions and then finding geometric series. Do I rewrite the denominator as 1-(z^3+1)? This one I'm totally confused on. Please help
 
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  • #2
You don't need partial fractions here. Think of a way to write the sine function as a series.
 
  • #3
Always remember that a Laurent series is unique, so if you can find a way to express a function as a power series, then it is the Laurent series.
 

1. What is a Laurent series?

A Laurent series is a representation of a complex function as an infinite sum of terms, including both positive and negative powers of the variable, centered around a specific point in the complex plane. It is similar to a Taylor series, but allows for the presence of singularities in the function.

2. How is a Laurent series different from a power series?

A power series only includes positive powers of the variable and is centered around a point where the function is analytic (meaning it has derivatives of all orders). A Laurent series, on the other hand, can include both positive and negative powers and may be centered around a point with singularities.

3. What is a singularity in a Laurent series?

A singularity is a point in the complex plane where the function is not analytic. This means that the function either has a pole (a point where it becomes infinitely large) or an essential singularity (a point where it cannot be expanded into a power series).

4. How do you find the coefficients in a Laurent series?

The coefficients in a Laurent series can be found using the formula cn = (1/2πi) ∮(f(z)/zn+1)dz, where the integral is taken around a closed contour in the complex plane containing the point of interest. Alternatively, the coefficients can also be found by expanding the function into a power series and then rearranging the terms to include both positive and negative powers.

5. What are partial fractions and how are they related to Laurent series?

Partial fractions are a way of decomposing a rational function into simpler fractions. They are useful for integration and solving differential equations. They are related to Laurent series because the coefficients in a Laurent series can also be found by using partial fractions to decompose the function into simpler terms.

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