Evaluating Complex Contour Integrals: (z+2)/z on the Top Half of a Circle

[bugatti79]

Homework Statement

Folks,

How do I evaluate the integral of (z+2)/z dz for the path C= the top half of the circle |z|=2 from z=2 to z=-2.

The Attempt at a Solution

I take $z=x+iy$ and $dz=dx+idy$

Therefore $\int_c f(z)=\int_c (1+(2/(x+iy))(dx+idy)$...not sure if I'm going the right direction

Or do I parameterise z as $z(t)=e^{it}$..?

Thanks

 

[Dick]

Parametrize as z=2e^(it). What is dz?

 

[bugatti79]

Parametrize as z=2e^(it). What is dz?

Thank you, sorted.

Stuck on this one. $\int_c (x^2+ixy) dz$ where C is given by $z(t)=t^2+t^3i$ for $0\le t\le1$

I thought of converting z to polar coordinates where $z=r\cos \theta + ir \sin \theta$ ad $x=r\cos \theta$ so we have

$\int_c x(x+iy)dz=\int r\cos\theta(r \cos \theta+i r\sin \theta)(-r\sin \theta d\theta+i r \cos \theta d \theta)$...the limits not sure how to approach, perhaps approach is wrong..?

 

[Dick]

Thank you, sorted.

Stuck on this one. $\int_c (x^2+ixy) dz$ where C is given by $z(t)=t^2+t^3i$ for $0\le t\le1$

I thought of converting z to polar coordinates where $z=r\cos \theta + ir \sin \theta$ ad $x=r\cos \theta$ so we have

$\int_c x(x+iy)dz=\int r\cos\theta(r \cos \theta+i r\sin \theta)(-r\sin \theta d\theta+i r \cos \theta d \theta)$...the limits not sure how to approach, perhaps approach is wrong..?

I don't think polar coordinates are any help. Just write it as an integral dt.

 

[bugatti79]

I don't think polar coordinates are any help. Just write it as an integral dt.

But how do I handle the x outside the bracket when we let z=(x+iy) inside the brackets?

 

[Dick]

But how do I handle the x outside the bracket when we let z=(x+iy) inside the brackets?

If z=t^2+it^3 then x=t^2 and y=t^3, right?

 

[bugatti79]

If z=t^2+it^3 then x=t^2 and y=t^3, right?

should have spotted that. thanks