# Math Genius: Finding all solutions \$z=x+iy\$ for four basic problems

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}

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