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Chrysostomou

Member

Registered: 2009-05-25

Posts: 9

complex integral

I do know what to do!Is someone know please help!!!:/

where C is a circle centred in a, unless n=-1.

around the following countours:
(i)the circle centred on 2,5 with radius 2,5
(ii)the circle centred on 4+5i with radius 7
(iii)the circle centred on 2 with radius 3/4
(iv)the circle centred on 3 with radius 01

Thanks Chrysostomou

Chrysostomou

Member

Registered: 2009-05-25

Posts: 9

Re: complex integral

I have just found out how to do the second question if someone know something about the first please tell me

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: complex integral

Parametrise C by f:[0,2π] -> C, where f(t) = a + ke[sup]it[/sup]. (k is some real constant, determined by the radius of the circle C)
Then:

This simplifies quite a lot and will equal 0 if n≠-1.

Edit: Just realised I was very ambiguous there. C is the circle, C is the set of complex numbers. Looks like you understood me anyway though.

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Why did the vector cross the road?
It wanted to be normal.

Chrysostomou

Member

Registered: 2009-05-25

Posts: 9

Re: complex integral

Many thanks!!
Can I ask something?
when sometimes the exercises ask to use the Cauchy integral theorem how is work!?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: complex integral

when sometimes the exercises ask to use the Cauchy integral theorem how is work!?

There are multiple versions of Cauchy's Integral Theorem, so you need to be more specific.  The simplest version says that if f is analytic in ball of radius R, then the integral of f around that ball is going to be 0.  Actually, it really says a bit more than that, it says that f has a primitive (and so the integral around any closed curve in that ball is zero).

But as this example demonstrates, f may very well have a primitive and the CIT need not be applicable.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Chrysostomou

Member

Registered: 2009-05-25

Posts: 9

Re: complex integral

here is some exercises I have:
use the Cauchy integral theorem to evaluate the following contour integrals:

where C is the unit circle |z|=1

or use Cauchy integral theorem to find the following contour integrals:

where C is the circle centred on z=5, radius 1 in the anticlockwise direction

Many thanks:)