/sci/ - Science & Math

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Ok this i what i found about the proof of the uniform convergence of the complex exponential function [math]\exp(z)[/math]. I want to know about its uniform convergence beacause it's needed to prove the Euler's formula [math]\exp(ix)=\cos(x)+i\sen(x)[/math] and that the Cauchy Product yields [math]\exp(zw)=\exp(z)+\exp(w)[/math].

I found that its pointwise convergence is not hard to show, using the ratio test for series of complex numbers. To show the ratio test one uses the geometric series convergence.

Next, one needs to show that is uniformly convergent and this is reached using the Weierstrass M-test. If one applies this test to [math]\exp(z)[\math] for z in bounded sets of the complex plane one finds the larger disk that contains such set and taking its radius, together with some inequalities it boils down to the ratio test on the resulting M-series. Thus the exponential function not only pointwise convergent in the entire complex plane but also uniformly convergent for every bounded set (how does this allow one to rearrange the series to show Euler's formula or find the Cauchy product)? .

I have to prove:
-Geometric series convergence
-Ratio test theorem
-Weierstrass M-test theorem

The stupid question that don't deserve its own thread is if this is the most elementary but rigorous proof for the full definition of the exponential function?