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Sorry about the confusing title but I'll try to illustrate what I mean here.

I have the equation $z=e^{-i(t+z)}$ where $z$ can be of the form $x+iy$, $x$, or $iy$ (for $x,y,t \in \mathbb{R}$). How do I convert this into rectangular form without splitting into cases? Or is this a necessity?

Edit:

I think I just simplified my question too much to put it up here. What I am trying to separate is this complex number:

z = exp$\left[-i(\omega t-k\left(\sin({\theta_{1}}) x + \sqrt{\sin^{2}{\theta_{2}}-\sin^2{\theta_{1}}}\right) \right]$

where I need to consider both $\theta_{1} < \theta_{2}$ and $\theta_{2} < \theta_{1}$ and $\omega t$ and $k\sin{\theta_{1}}$ are both real.

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  • $\begingroup$ Strictly speaking $e^{-i(t+z)}$ is not an equation $\endgroup$ – Henry Sep 16 '15 at 23:52
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Well if $z$ can be complex then:

$$e^{-i(x+iy+t)}=e^{-ix+y-it}=e^{-ix}e^{y}e^{-it}$$

You know $x,y,t$ are all real. Just use Euler's formula $$e^{i\theta}=\cos\theta+i\sin\theta$$ and you'll get it split into cartesian form.

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  • $\begingroup$ Ah I see what you mean, I think I just simplified my question too much to put it up here. What I am trying to separate is this complex number: exp$\left[-i(\omega t-k\left(\sin({\theta_{1}}) x + \sqrt{\sin^{2}{\theta_{2}}-\sin^2{\theta_{1}}}\right) \right]$ where I need to consider both $\theta_{1} < \theta_{2}$ and $\theta_{2} < \theta_{1}$. Does your answer still apply? $\endgroup$ – Daniel Ward Sep 16 '15 at 23:52
  • $\begingroup$ Also $\omega t$ and $k\sin{\theta_{1}}$ are both real. $\endgroup$ – Daniel Ward Sep 16 '15 at 23:55
  • $\begingroup$ Which of your quantities can be complex? $\endgroup$ – DLV Sep 17 '15 at 0:17
  • $\begingroup$ In general just use exponent rules, and then Euler's formula to convert to cartesian form. $\endgroup$ – DLV Sep 17 '15 at 0:17

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