# Thread: Complex Analysis: Finding Roots

1. ## Complex Analysis: Finding Roots

Find all roots tanh(z)=i

tanhz = [e^(z)-e^(-z)]/[e^(z)+e^(-z)]

where do i go from here?

2. Originally Posted by CarmineCortez
Find all roots tanh(z)=i

tanhz = [e^(z)-e^(-z)]/[e^(z)+e^(-z)]

where do i go from here?
Here's a start. Re-arrange:

$e^z - e^{-z} = i \left( e^z + e^{-z}\right)$

$\Rightarrow (1 - i) e^z = (1 + i) e^{-z}$

$\Rightarrow e^{2z} = \frac{1 + i}{1 - i} = i$.

3. Originally Posted by mr fantastic
Here's a start. Re-arrange:

$e^z - e^{-z} = i \left( e^z + e^{-z}\right)$

$\Rightarrow (1 - i) e^z = (1 + i) e^{-z}$

$\Rightarrow e^{2z} = \frac{1 + i}{1 - i} = i$.
I don't understand the factoring in you second line, what is that?

4. Originally Posted by CarmineCortez
I don't understand the factoring in you second line, what is that?
At this level of mathematical studies you should have seen the following steps (*):

$e^z - e^{-z} = i \left( e^z + e^{-z}\right)$

* $\Rightarrow e^z - e^{-z} = i e^z + i e^{-z}$

* $\Rightarrow e^z - i e^z = e^{-z} + i e^{-z}$

$\Rightarrow (1 - i) e^z = (1 + i) e^{-z}$