For each of the following expressions find all such that the formula holds.

- .
- .
- .
- .

- We use the definition of the complex exponential we compute,
Then we set the real parts and the imaginary parts on each side equal to each other,

This implies and is arbitrary. (Since implies which is only true for . But then, for all ; hence, we must have from the second equation. If then both equations hold for all .)

- Similarly to part (b) we have,
Setting real and imaginary parts equal we have,

This implies .

- Again, we compute
Therefore,

This implies

But, this implies and so .

- Finally,
Setting real and imaginary parts equal we have

This implies

For d) the last system of equations is leading to a wrong result. The correct solution is to divide the first equation by the second (which is nonzero) to obtain:

Therefore,

for d) x=-1 and y=3Pi/2 + 2pi is also a solution

I agree.