Integral Question

IsThisIt???

Hey, would anyone be able to help me out with this Integral question. I think it's a pretty long answer so if someone could give me a pointer on where to get started that would be great. I don't even know what my first move is.

The question is evaluate the following

Eliot Rosewater

Is this from a complex analysis problem set? I would evaluate this using complex analytic methods, such as the Cauchy Residue Theorem. Is this a suitable line of approach?

Michael Collins

If it is a complex variable problem, which is looks to be, check out a similar problem and solution in this post: http://www.boards.ie/vbulletin/showpost.php?p=68782941&postcount=18

IsThisIt???

:eek:

....okay, thanks for the link,think I've got some studying of this to do but that answer helps a lot.

This was a question from an exam paper a few years ago for my analysis class.

sponsoredwalk

By Chebyshev's theorem the indefinite integral is expressible in terms of elementary functions. Furthermore you can factor this either by finding the roots of the polynomial x⁴ + 1, which amounts to finding the complex roots of -1 & since you've got complex roots of a polynomial w/ real coefficients they come in conjugate pairs which you can use to get rid of the complex numbers, or you can try to turn it into the difference of two squares & then factor that:
x⁴ + 1 = (x²)² + 1² = (x²)² + 2x² + 1² - 2x² = ((x²) + 1)² - 2x² = (x² + 1 + √2x)(x² + 1 - √2x)

thus you'll end up with a product of two quadratics in the denominator. After that it'll be alright I'd say, though you might need some weapons...

IsThisIt???

Made some progress with that one, but I'm still working on it.

another integral question here from the same paper, don't think it's similar to the one I already posted but I really don't know.

Find all the possible values of


where y(the curve) is a simple closed curve in C oriented counterclockwise.

I think I'm supposed to use the Cauchy Integral Formula but it seems like some information is missing. Could anyone point me in the right direction??

(Sorry for all the questions lately, exam is tomorrow and really trying to cover all my bases)

Michael Collins

Yeh, you need to know the nature of the curve [latex] \gamma [/latex].

Say, for example, it is a circle centred on the origin (0,0) with radius 1. Then you would use Cauchy's Integral Formula, and note that there are two simple poles one at z=0, another at z=-100i, and only the former is inside the example curve I've given, so that's the one to use as the denominator of Cauchy's Integral Formula. This then says

[latex] \displaystyle \oint_{\gamma} \frac{f(z)}{z-a} \hbox{d}z=2\pi \hbox{i}f(a) [/latex]

where in your case f(z) would be

[latex] \displaystyle f(z) = \frac{\cos(z+1)}{z+100\hbox{i}} [/latex]

so f(a) and the answer should follow fairly easily.