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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

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2 Answers

up vote 2 down vote accepted

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
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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 $e^{4x}\sin{4y}=1$.

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@Chandru1, typo: missing $i$ in $\sin 4y+i\cos 4y$. –  Américo Tavares Jan 6 '11 at 17:14
    
@maerico: thanks –  anonymous Jan 6 '11 at 17:16
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