What tools are you allowed to use? Cauchy's integral theorem? Residue theorem?
Hello,
I have been working on complex integration in my own time and have managed to solve a few, but these three seriously have me stumped. I was wondering if anyone would be willing to help me understand these problems.
a) where is the circle of radius 4 about the origin, traversed twice in the clockwise direction.
b) where proceeds around the boundary of the figure eight formed by two circles of radius 1 with centres 1 and −1 by starting at 0, going once counterclockwise around the right circle followed by going once counterclockwise around the left circle.
c) where is the circle of radius 2 about the origin.
I thank you all in advance! Your help is very much appreciated.
Hey, sorry for the delay.
For your first post - yes they will be very useful here.
What is absolutely essential here is that you draw the curves that you are integrating over, so that you know how to parameterize them. This is important not because you will use the parametrizations explicitly, but it will give you an idea of how to use the needed theorems.
I'll do the first one for you - try to do the others in a similar way.
Let where is as mentioned.
Since this is not a simple curve (it intersects itself) you will have to reparameterize it as a concatenation of two simple curves where traverses once, clock-wise.
By properties of concatenation of curves, you have
Now, using the residue theorem:
where and are the poles of inside -
Now calculate the residues:
And so
And finally
If you are not familiar with all of this then you are probably expected to use Cauchy's integral theorem. If you need that I'll also show you how to solve this integral with that.