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**Test 1 / 2020S-TT1 Q1**

« **on:**October 14, 2020, 03:42:15 PM »

I'm having trouble understanding where the $-1+i$ term comes from in the following line:

$\dfrac{e^{3z} - e^{-3z}}{e^{3z} + e^{-3z}} = 1 + 2i \implies e^{6z} = -1 + i$.

I've tried the following:

$$\begin{align*}

\dfrac{e^{3z} - e^{-3z}}{e^{3z} + e^{-3z}} &= 1 + 2i\\

\dfrac{e^{6z} - 1}{e^{6z} + 1} &= 1 + 2i\\

e^{6z} - 1 &= (1 + 2i)(e^{6z} + 1)\\

e^{6z} &= (1 + 2i)(e^{6z} + 1) + 1

\end{align*}$$

How do we get from $(1 + 2i)(e^{6z} + 1) + 1$ to $-1 + i$? Have I done something wrong somewhere in my calculation?

$\dfrac{e^{3z} - e^{-3z}}{e^{3z} + e^{-3z}} = 1 + 2i \implies e^{6z} = -1 + i$.

I've tried the following:

$$\begin{align*}

\dfrac{e^{3z} - e^{-3z}}{e^{3z} + e^{-3z}} &= 1 + 2i\\

\dfrac{e^{6z} - 1}{e^{6z} + 1} &= 1 + 2i\\

e^{6z} - 1 &= (1 + 2i)(e^{6z} + 1)\\

e^{6z} &= (1 + 2i)(e^{6z} + 1) + 1

\end{align*}$$

How do we get from $(1 + 2i)(e^{6z} + 1) + 1$ to $-1 + i$? Have I done something wrong somewhere in my calculation?