1. ## Complex trig equation

The question
Find all solutions to $z \in C$ of:
cosh(z) = 2i

My attempt
$cosh(z) = \frac{e^z + e^{-z}}{2} = 2i$

$e^z + e^{-z} = 4i$

$e^{2z} + 1 = 4ie^z$ //multiply by $e^z$

$e^{2z} - 4ie^z = -1$

$(e^z - 2i)^2 + 4 = -1$ //complete the square

$e^z - 2i = \sqrt{-5}$

$e^z = \sqrt{-5} + 2i$

$e^z = i(\sqrt{5} + 2)$

$e^xe^{iy} = (\sqrt{5} + 2)e^{\frac{\pi}{2}i}$

so $e^x = \sqrt{5} + 2$

$x = ln(\sqrt{5} + 2)$

$y = \frac{\pi}{2} + 2k\pi$ //since exp is $2k\pi i$periodic

Since z = x + iy, $z = ln(\sqrt{5} + 2) + i\pi(\frac{1}{2} + 2k)$

However, the solution is actually:
$(-1)^nsinh^{-1}(2) + (n + \frac{1}{2})\pi i$

I'm confused, my working looks correct. :/ Any help would be greatly appreciated!

2. ## Re: Complex trig equation

Originally Posted by Glitch
The question
Find all solutions to $z \in C$ of:
cosh(z) = 2i

My attempt
$\cosh(z) = \frac{e^z + e^{-z}}{2} = 2i$

$e^z + e^{-z} = 4i$

$e^{2z} + 1 = 4ie^z$ //multiply by $e^z$

$e^{2z} - 4ie^z = -1$

$(e^z - 2i)^2 + 4 = -1$ //complete the square

$e^z - 2i = \sqrt{-5}$ Should be $\color{red}\pm\sqrt{-5}$.

$e^z = \sqrt{-5} + 2i$

$e^z = i(\sqrt{5} + 2)$

$e^xe^{iy} = (\sqrt{5} + 2)e^{\frac{\pi}{2}i}$

so $e^x = \sqrt{5} + 2$

$x = \ln(\sqrt{5} + 2)$

$y = \frac{\pi}{2} + 2k\pi$ //since exp is $2k\pi i$-periodic

Since z = x + iy, $z = \ln(\sqrt{5} + 2) + i\pi(\frac{1}{2} + 2k)$

However, the solution is actually:
$(-1)^n\sinh^{-1}(2) + (n + \frac{1}{2})\pi i$

I'm confused, my working looks correct. :/ Any help would be greatly appreciated!
Your answer is (half) correct, because $\sinh^{-1}2 = \ln(\sqrt 5 + 2)$. See here, for example. You can get the other half of the answer by following through what happens if you use $-\sqrt{-5}$ instead of $\sqrt{-5}.$

3. ## Re: Complex trig equation

You should know that $\displaystyle \cosh{(z)} = \cosh{(x + iy)} = \cosh{(x)}\cos{(y)} + i\sinh{(x)}\sin{(y)}$.

So you are solving $\displaystyle \cosh{(x)}\cos{(y)} + i\sinh{(x)}\sin{(y)} = 2i$.

Equating real and imaginary parts gives $\displaystyle \cosh{(x)}\cos{(y)} = 0$ and $\displaystyle \sinh{(x)}\sin{(y)} = 2$.

Solving the first equation: Since $\displaystyle \cosh{(x)} > 0$ for all $\displaystyle x$, that means $\displaystyle \cos{(y)} = 0 \implies y = \frac{(2n + 1)\pi}{2}$, where $\displaystyle n \in \mathbf{Z}$.

Substituting into the second equation $\displaystyle \sinh{(x)}\sin{(y)} = 2$ gives

\displaystyle \begin{align*}\sinh{(x)}\sin{\left[\frac{(2n + 1)\pi}{2}\right]} &= 2 \\ (-1)^{n}\sinh{(x)} &= 2 \\ \sinh{(x)} &= \frac{2}{(-1)^n} \\ x &= \sinh^{-1}{\left[\frac{2}{(-1)^n}\right]} \end{align*}

So the solution is $\displaystyle z = \sinh^{-1}{\left[\frac{2}{(-1)^n}\right]} + \frac{(2n + 1)\pi i}{2}$, where $\displaystyle n \in \mathbf{Z}$.