Several problems on series representations, residue theorem

[ColdFusion85]

First question pertains to the Residue Theorem

We are to use this theorem to evaluate the integral over the given path...

There is one problem from this section that I am stuck on. An example in the book evaluates

\int_{\Gamma} e^{1/z} dz for \Gamma any closed path not passing through the origin.

We need Res(e^{1/z}, 0)

It was found in a previous example that 0 is an essential singularity of e^{1/z}. There is no simple general formula for the residue of a function at an essential singularity. However,

e^{1/z} = \sum_{n=0}^\infty \frac{1}{n!}\frac{1}{z^n}

is the Laurent expansion of e^{1/z} about 0, and the coefficient of 1/z is 1. Thus, Res(e^{1/z},0) = 1 and

\int_{\Gamma} e^{1/z} dz = i2\pi
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So, my problem is \int_{\Gamma} e^{\frac{2}{z^2}} dz, and gamma is the square with sides parallel to the axes and of length 3, centered at -i

What I did was say that e^\frac{2}{z^2} = \sum_{n=0}^\infty \frac{1}{n!}\frac{2}{z^{2n}}

and again 0 is an essential singularity of the function. The coefficient of \frac{2}{z^2} is 2, and therefore Res(e^{\frac{2}{z^2}},0) = 2, and so
\int_{\Gamma} e^\frac{2}{z^2} dz = i4\pi

Is this correct?

 

[ColdFusion85]

Second problem from a section on Laurent Expansions. All of the problems in the book deal with functions that have a constant in the numerator [e.g., 1/(2+z)] or that we know the expansion of already [e.g., sin(z)/z]

However, the problem I am stuck on from this section is

\frac{z^2 + 1}{2z-1} about 1/2

How do I get the expansion of such an equation when the variable is both in the numerator and denominator, and there is no clear expansion that we know of (i.e. for z^2 +1)?

Finally, a problem from a section on singularities. We are to determine all singularities of the function and classify each as removable, a pole of a certain order, or an essential singularity.

The function I have problem on is

\frac{4sin(z+2)}{(z+i)^2(z-i)}

The singularities are +/- i, but how do I simplify or expand this equation out to determine what the classification of these singularities are?

Thanks for any help you can give me on any or all of these three problems!

 

[ColdFusion85]

Anyone?