# Homework Help: Complex numbers with trig functions

1. Apr 4, 2009

### thanksie037

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

prove: arctanh(z) = 1/2 ln( (1+z)/(1-z) )

2. Relevant equations

cosh z = (ez + e-z)/2
sinh z = (ez - e-z)/2
ez = ex + iy = ex(cosy + siniy)

3. The attempt at a solution

cosh z / sin hz = ez+e-z/ez-e-z

2. Apr 4, 2009

### Dick

tanh(z)=sinh(z)/cosh(z). Not cosh(z)/sinh(z). This is just like the problem of finding the inverse of y=f(x). First you try and solve for x in terms of y then interchange x and y. Try and solve y=(e^x-e^(-x))/(e^x+e^(-x)) for x in terms of y and then interchange x and y. Hint: multiply numerator and denominator by e^x.

3. Apr 4, 2009

### thanksie037

sorry, the problem specifies artanh . . .
solving for e^x wont give me the same properties that using z implies (since z = xi + y)

4. Apr 4, 2009

### Dick

The inverse function of y=tanh(z) is arctanh. You don't need to break z down to it's real and imaginary components. You need to solve that equation for z in terms of y and then replace y with z.

5. Apr 7, 2009

### thanksie037

thanks! I finally got it. i appreiate your help. cool problem.