1. ## some complex variables

As you can see from the checkmarks I already got 6a and 4
If someone could just help me with the notation on 1, like what do those (+) things mean?

2. I'm pretty sure that's just a polygonal path (straight-line) path from the points $1,i,-1,-i,1$. Looks pretty messy too.
If I calculate an antiderivative:

$\int_0^{1}\frac{1}{z+t}dt=\text{Log}\left(\frac{z+ 1}{z}\right)$

and then use this to calculate the other integral, I get for the first leg:

$\mathop\int\limits_{L_1}\frac{1}{z}dz=\frac{-1+i}{i-1}\text{Log}\left(\frac{\frac{1}{i-1}+1}{\frac{1}{i-1}}\right)=\pi/2 i$

Maybe easier way though.

3. well one thing i just thought of is, this is a closed piecewise smooth curve so wouldn't the integral be 0?

4. Originally Posted by jbpellerin
well one thing i just thought of is, this is a closed piecewise smooth curve so wouldn't the integral be 0?
No. The integral is zero only if the integrand is analytic in and on a closed contour and 1/z is not in any closed contour about the origin.