# Calculation residues at z=1 of order 4

• Kowsi Ram
Thank you for the tip!In summary, the residue of the given function at z=1 can be calculated by expanding it into a Laurent series, rather than differentiating it multiple times. This can be done by multiplying geometric series together.
Kowsi Ram

## Homework Statement

Find the residue of the following function at z=1

## Homework Equations

f(z)=z^2/(z-1)^4(z-2)(z-3)

## The Attempt at a Solution

lim z→2 z^2/(z-1)^4(z-3)=-4

limz→3 z^2/(z-1)^4(z-2)=9/16

at z=1 of order 4?? using formula 1/m-1! d/dz{(z-a)^m f(z)}

Kowsi Ram said:

## Homework Statement

Find the residue of the following function at z=1

## Homework Equations

f(z)=z^2/(z-1)^4(z-2)(z-3)

## The Attempt at a Solution

lim z→2 z^2/(z-1)^4(z-3)=-4

limz→3 z^2/(z-1)^4(z-2)=9/16

at z=1 of order 4?? using formula 1/m-1! d/dz{(z-a)^m f(z)}

Looks good, except that you need to take m-1 derivatives (why?).

I think it would be better if you did your homework yourself.

Remember the definition of a residue -- there's an easier way to calculate it in this case, than differentiating the function three times. Expand it into a Laurent series around z=1 by multiplying a couple of geometric series together.

You are correct. I will try this method.

## What does it mean to calculate residues at z=1 of order 4?

Calculating residues at z=1 of order 4 means finding the values of the function at z=1 and its first three derivatives, and then using those values to determine the coefficient of the (z-1)^4 term in the Laurent series of the function.

## Why is it important to calculate residues at z=1 of order 4?

Calculating residues at z=1 of order 4 is important because it allows us to determine the nature of singularities at z=1. This information is crucial in understanding the behavior of the function near z=1 and making accurate calculations and predictions.

## How do you calculate residues at z=1 of order 4?

To calculate residues at z=1 of order 4, you first need to find the values of the function and its first three derivatives at z=1. Then, you can use the formula Res(z=1) = (1/3!) * (d^3/dz^3)[(z-1)^4 * f(z)]|z=1 to determine the coefficient of the (z-1)^4 term in the Laurent series.

## Can you calculate residues at z=1 of order 4 for any function?

Yes, you can calculate residues at z=1 of order 4 for any function that is analytic at z=1. This means that the function must be differentiable an infinite number of times at z=1.

## How are residues at z=1 of order 4 used in real-world applications?

Residues at z=1 of order 4 are used in many areas of mathematics and science, such as complex analysis, differential equations, and physics. They can help us understand the behavior of functions near singularities and make accurate predictions and calculations in various real-world scenarios.

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