# Complex Analysis help?

1. May 8, 2014

### 159753

1. The problem statement, all variables and given/known data

I can't seem to get a few questions involving inverse trigonometric functions and hyperbolic functions. Here is one that I am stuck on:

Evaluate the following in the form x+iy:

sinh-1(i/2) = z

2. Relevant equations

sinh z = (ez - e-z)/2

3. The attempt at a solution

sinh-1(i/2) = z
sinh (z) = i/2

This means that i/2 = (ez - e-z)/2

Let u = ez

Where do I go from here? I don't know how to deal with the imaginary number.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 8, 2014

### xiavatar

Actually there is a simpler way. Note that $sinh^{-1}(z)=ln(z+\sqrt{z^2+1})$

Last edited: May 8, 2014
3. May 8, 2014

### 159753

I got the answer. To get the solution, I put all the terms on one side and used the quadratic formula to find the solutions of u. From there, it is easy enough to figure out.

The final answers are i(Pi/6 +2nPi); i(5Pi/6 +2nPi)

4. May 8, 2014