Branches of analytic functions 
Find a branch of $\log(z^2+1)$ that is analytic at $z=0$ and takes the
  value of $2\pi i$ there. Also, determine a branch of $\log(z^2+2z+3)$ that is analytic at $z=-1$.

If I plug in $z=0$ and $z=-1$ to there respective functions then I get $\log(1)$ and $\log(2)$ but then what do I have to do to find a branch? 

Define a branch of $(z^2-1)^{1/2}$ that is analytic in the exterior of
  the unit circle $|z|>1$.

If I transform this function into $e^{(1/2)Log(z^2-1)}$, why will the principal branch not work here? 

Let me give a simpler function so you can see what's going on. Let's consider the function $f(z)=\ln(z)$. We would like to find a branch where $\ln(z)$ is analytic at $z=1$ and has a value $2\pi i$ there. Now, you know that
$$ \ln(z)=\ln|z|+i(\theta+2k\pi) \longrightarrow  (1). $$
We want to find the branch of the function that has value $2\pi i$ at 
$$z=1 \implies z=e^{0i} \implies \theta = 0 . $$
Substituting $\theta =0 $ in $(1)$, $z=1$, and choosing the right branch, in our case $k=1$, we have
$$ \ln(1)= 0 + i(0+2\pi)=2\pi i. $$
I hope this example makes it clear for you. 
