ComplexAnalysis tan z

1. Mar 18, 2005

Mark C

Hi

If w=tan z

How would I show that any w not equal to +i or -i,
is the image of some z in C, and what are the solutions
of w=tan z if w doesnt equal to +i or -i?
I can easily show that tan z can never equal +i or -i,
but thats not the same thing also.

Note: without using the arctan w identity.

Thank you

2. Mar 18, 2005

matt grime

tan is sin over cos, right? and these are simply exponentials in z, too. does that help?

3. Mar 18, 2005

Mark C

Well, yes I know that, I can show that if w=i or -i then the expression is not defined, but thats not the same thing.

thank you anyway

4. Mar 18, 2005

hypermorphism

Have you derived the inverse function using the exponentials ?

5. Mar 18, 2005

Mark C

I am not supposed to use the formula for arctan w.

6. Mar 18, 2005

mathwonk

tanz is the same as e^z but translated so that i , -i correspond to 0 and infinity. maybe. or you could use picards theorem.

Last edited: Mar 18, 2005
7. Mar 19, 2005

matt grime

Deriving it isnt' the same as "using it" you simply show that there is no solution in terms of the exponential version of the functions.

8. Mar 19, 2005

mathwonk

here is a quick argument for you.
fact: the unit disc is the universal covering space of: "the sphere minus 3 points".

corollary: if a holomorphic map to the sphere misses three points, then it factors through the unit disc, hence is constant, by liouville's theorem.

since tan(z) is not constant it cannot miss any more than the two points i and -i.