Hey flametag3.

If these functions are analytic then they will be finite everywhere where they are analytic which means the modulus will be < infinity.

So recall that the modulus square can be calculated by doing z * z_bar for a complex number z so from this hint, one way is calculate the modulus of the function mapping (which is just a complex variable and subsequently just a complex number).

Another way is by examination showing that for example e^z is defined everywhere for all complex z and so is e^z + 1 and square root of this is defined everywhere, while 1/(e^z - 1) is not defined at z = 0.

You can show this with the modulus where the modulus is always less than infinity as long as z is not infinity but for the second one if z = 0 then the modulus blows up to infinity.