Complex integrals along line segment and semicircle contours

[LiamH]

Homework Statement

Hey people got a question here about complex integration, not really sure how to do it so hope someone out there could help me!

Evaluate the complex integrals

∫ c { (zbar)^2 +1 } dz...and...∫ c { zcos(z^2) - ie^2z }

where c is the contour joining 0 to 2i along
(i) the line segment parametrized by z(t) = 2it, t∈{0,1}
(ii) the semicircle parametrized by z(t) = i + e^it, t∈{-pi/2, pi/2}

Homework Equations

The Attempt at a Solution

I know we have to sub the parameterization in for z, but now that there's a line and semicircle I am confused! Didnt have a great prof for this subject, so I am hopin some other people could help me out

 

[lanedance]

hi liamH

as you are given the parameterisaton, why not try substituting it into the inetrgal, then the integration should become a simple single variable integral. Remember you will have to account for the dz as well.

 

[LiamH]

Ok so for the first integral and the line segment...

int from 0 to 1 {(2it bar) ^ 2 +1} (2i) dz

i worked this out and the answer i got was -2/3 i?

is this right?

and so for the semicircle the integral wil be

int from -pi/2 to pi/2 {(-1 + e^it bar) ^2 + 1 (e^it) } ?

 

[lanedance]

first one looks correct - probably easier if you show you working and I can check it rather than me doing the whole integral as well