Contour integral, taylor and residue theory question

In summary, the conversation discusses using the Exponential Taylor Series and the Residual Theorem to find the Laurent Series form of a given equation involving cosine and inverse z. The solution involves using the Taylor's series for cosine and manipulating it to get a power series in negative powers of z, which is then integrated to find the desired term. The relevance of the Taylor's series for e^z is also mentioned in the conversation.
  • #1
Ian_Brooks
129
0

Homework Statement



http://img243.imageshack.us/img243/4339/69855059.jpg [Broken]
I can't seem to get far. It makes use of the Exponentional Taylor Series:

Homework Equations



http://img31.imageshack.us/img31/6163/37267605.jpg [Broken]


The Attempt at a Solution


taylor series expansions for cos and 1/z i assume - and stick it into the residual theorem, but i need to get it into the laurent series form first so that i can find out b1. I think it's like pole order 4 or 5 so it's going to be a pain
 
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  • #2
my attempt at a solution
is this on the right track?
http://img219.imageshack.us/img219/2425/image123t.jpg [Broken]
 
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  • #3
It has been a while but it does look correct to me. You can do it a little bit easier though if you know the series expansion of the cosine.

[tex]
z^5 \cos(\frac{1}{z})=z^5 \sum_{k=0}^{\infty} (-1)^k \frac{\left(\frac{1}{z}\right)^{2k}}{(2k)!}=...+\frac{-1}{6!z}+...
[/tex]
 
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  • #4
Why is the Taylor's series for ez a "relevant" equation? Use the Taylor's series for cos(z), replace z by 1/z so that you get a power series in negative powers of z (a "Laurent series") and then multiply by [itex]z^5[/itex] so you have a power series with highest power z5 and decreasing. It should be easy to see that, integrating term by term, every term gives 0 except the z-1 term.
 

1. What is a contour integral?

A contour integral is a type of complex integral that is evaluated along a specific path or contour in the complex plane. It is used to calculate the area under the curve of a complex function.

2. How is the residue of a function calculated?

The residue of a function is calculated by finding the coefficient of the term with a negative power in the Laurent series expansion of the function. This coefficient is also known as the residue of the function at a specific point.

3. What is Taylor series and how is it related to contour integrals?

Taylor series is a representation of a function as an infinite sum of terms, each with a higher degree of the independent variable. It is related to contour integrals as the coefficients in the Taylor series can be used to calculate the values of the function at different points, which is useful in evaluating contour integrals.

4. What is the Cauchy integral formula?

The Cauchy integral formula states that if a function is analytic (differentiable) within a closed contour, then the integral of the function along that contour is equal to the sum of the residues of the function at all points inside the contour.

5. How are contour integrals used in real-world applications?

Contour integrals are used in many areas of science and engineering, such as in electrical engineering to calculate the electric field around a charged object, in physics to calculate the work done by a force on an object, and in economics to calculate the total revenue from a production function.

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