I am preparing for an exam on Complex Analysis and I have four small problems. Two of them I think I have solved but I would really appreciate a sanity check since I very often miss something. On the other two, I have no idea how to start, any hint/suggestion would be appreciated. I need to find **all** solutions $z=x+iy$.

$textbf{a) } z+1 = log{(i+1)}$. The typesetting won’t allow it, but is is actually Log, hence the principal branch.

$begin{align}

z+1 &= log{i+1}\

z+1 &= ln{sqrt{2}} + frac{pi}{4}\

z &= ln{sqrt{2}} + frac{pi}{4} -1

end{align}

$

$textbf{b) } sin{z} = -i$. My solution is

$begin{align}

frac{e^{iz}-e^{-iz}}{2i} &= -i\

e^{iz}-e^{-iz} &= 2\

e^{iz}-e^{-iz} -2 &= 0 text{ , now let $a = e^{iz}$}\

a^{2}-2a-1 &= 0\

a &= 1 pm sqrt{2}

e^{iz} = 1 pm sqrt{2}\

iz &= ln{(1 pm sqrt{2})}\

z &= -iln{(1 pm sqrt{2})}.

end{align}

$

Now the two problems I do not know where to start:

$textbf{c) } (z+i)^{3} = 8. $ Writing it out does not seem to help me get anywhere, I then find $z^{3}-3z+3iz^{2}-i=8$.

$textbf{d) } (z+1)^{2i} =1$. No clue.

The first two are correct.

**c**) $(z+i)^3=8iff z+i=2vee z+i=2left(-frac12pmfrac{sqrt3}2iright)$

**d**) I assume that $z^w$ is defined as $expleft(woperatorname{Log}(z)right)$. If so,begin{align}(z+1)^{2i}=1&iffexpleft(2ioperatorname{Log}(z+1)right)=1\&iff 2ioperatorname{Log}(z+1)=2pi intext{ (for some integer }ntext{)}\&iffoperatorname{Log}(z+1)=pi n\&iff z+1=e^{pi n}.end{align}