Evaluating the argument of exp(4z)=i

Question

I am trying to find the complex roots of: exp(4z)=i.

However I am confused as how to calculate tan-1(1/0) to find the argument.

Any help would be much appreciated. Thanks

Answer 1

Two suggestions:

• consider for what value(s) of $x$ the expression $\tan x$ is undefined (which could correspond to your $\frac{1}{0}$)
• rather than applying a formula ($\tan\frac{y}{x}$, or something like that), think about where $i$ is located, specifically at what magnitude of rotation about $0$ from the positive real axis

Answer 2

So let $z=x+iy$. Then you have $4z=4x+i4y$. Then $$e^{4z}=e^{4x+i4y}=e^{4x}\Bigl[i\sin{4y} + \cos{4y}\Bigr]$$

Since this is you want $e^{4x}\cos{4y}=0$ and $e^{4x}\sin{4y}=1$.