(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve [tex]cosz = 2i , z\in \mathbb{C} [/tex]

3. The attempt at a solution

[tex]e^{iz}+e^{-iz} = 4i[/tex]

[tex]t=e^{-z} [/tex]

[tex]t+t^{-1}=4i \Rightarrow t^{2}-4it+1=0 [/tex]

[tex]t = (2 \pm \sqrt{5})i[/tex]

[tex]log(e^{-z}) = logt [/tex]

[tex]z = x + yi;x,y \in \mathbb{R}[/tex]

[tex]log(e^{-z}) = log(e^{-y+ix}) = -y +xi + 2\pi ni; n\in\mathbb{Z}[/tex]

[tex]logt = log((2 \pm \sqrt{5})i) = ln|(2 \pm \sqrt{5})| \pm \frac{\pi}{2} + 2\pi hi, h \in \mathbb{Z}[/tex]

[tex]\Rightarrow y= -ln|(2 \pm \sqrt{5})| [/tex]

[tex]\Rightarrow x= \pm \frac{\pi}{2} + 2\pi p, p \in \mathbb{Z}[/tex]

[tex]z= \pm \frac{\pi}{2} + 2\pi p -iln|(2 \pm \sqrt{5})|[/tex]

answer in book:

[tex]z = \pm\lbrace \frac{\pi}{2} -iln(2 + \sqrt{5})\rbrace+2\pi n, n\in \mathbb{Z}

[/tex]

Were did I do wrong? :S

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# Homework Help: Complex cosine equation (complex analysis)

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