How does this "PROOF" proves the theorem?

[X]

This theorem is from Apostol's Mathematical Analysis (2nd ed.).

I can't understand the reasoning behind the proof.



Please help.

Thank You.

[greenworld666]

value of e^1 is 2.718...
in the question if z>0 then the value of e^z will always increase
and if z<0 then it will always decrease
in both of the cases it is obvious that it wont be zero
only case left is e^0   i.e. when z=0
and we know that any number raise to zero is 1
and (e^z)* (e^-z) =e^(z-z)=e^0=1
therefore e^z cannot be zero.

[X]

Note that this theorem is about the complex exponential.(Here z is a complex number of the form a+ib where a and b are real numbers and i=((-1)^0.5).)

Note that for complex numbers you don't have ordering "facility"!!
So, you can't speak here about e^z>0 or e^z<0 .

Can anyone give other(and true) reasoning behind the proof?

The e-book is here:Apostol, Mathematical Analysis

Thank You.

[X]

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