ComplexAnalysis tan z

[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

[matt grime]

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

[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

[hypermorphism]

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
Have you derived the inverse function using the exponentials ?

[Mark C]

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

[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.

[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.

[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.