# Contour Integration help with understanding an example

• Dec 30th 2010, 03:12 AM
cooltowns
Contour Integration help with understanding an example
Hi

I have attached an example from my notes for contour integration.

However I don't understand how they derive the parametrized equation $\displaystyle \gamma(t)=t^{2}=it$

I would appreciate if someone could please explain this to me

Thanks
• Dec 30th 2010, 03:59 AM
chisigma
Take into account that is $\displaystyle z= x + i\ y = t^{2} + i\ t$ so that...

$\displaystyle \displaystyle |z|^{2} = x^{2} + y^{2} = t^{4} + t^{2}$ (1)

$\displaystyle \displaystyle dz = dx + i\ dy = (2\ t+ i)\ dt$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Dec 30th 2010, 06:00 AM
Ackbeet
If you're wondering how they came up with $\displaystyle \gamma(t)=t^{2}+it,$ I would say that they wrote the example that way. There's nothing to derive with respect to the choice of contour. There's loads of freedom in deciding on contours over which to integrate, depending on the situation.
• Dec 30th 2010, 06:16 AM
chisigma
It is important to note that $\displaystyle |z|^{2}$ isn't analytic so that the integral $\displaystyle \displaystyle \int_{0}^{1+i} |z|^{2}\ dz$ does depend from the path $\displaystyle \gamma$ connecting the point $\displaystyle a=0$ and $\displaystyle b=1+i$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Dec 30th 2010, 08:57 PM
SammyS
Quote:

Originally Posted by cooltowns
Hi

I have attached an example from my notes for contour integration.

However I don't understand how they derive the parametrized equation $\displaystyle \gamma(t)=t^{2}+it$

I would appreciate if someone could please explain this to me

Thanks

I can't tell you how they get the parametrized equation:

$\displaystyle \gamma(t)=t^{2}=it,\ \ 0\le t\le 1\ ,$

but I can tell you the path it describes.

$\displaystyle \displaystyle x+i\,y = t^2+i\,t$

then $\displaystyle \displaystyle y = t\,,$ and $\displaystyle \displaystyle x = t^2\,.$

thus: $\displaystyle \displaystyle x = y^2\,,$

Since $\displaystyle \displaystyle 0\le t\le 1\ ,$ we have $\displaystyle \displaystyle y = \sqrt{x}\,,\ 0\le x\le 1\,.$

The path is from point (0, 0) to point (1, 1) via $\displaystyle \displaystyle y = \sqrt{x}\,.$

Writing that in complex terms: The path is from z=0+0i to z=1+i via $\displaystyle \displaystyle z = x+i\sqrt{x}\,.$