Complex Analysis: Inverse Trig and Hyperbolic Functions Help

[159753]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

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

Let u = e^z

Where do I go from here? I don't know how to deal with the imaginary number.

 

[xiavatar]

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

 

[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)

 

[xiavatar]

Alright glad to hear that you got the answers.