User talk:Shahpour

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 * --Your friendly ProofWiki WelcomeBot 14:00, 4 May 2013 (UTC)

Please don't put work into random talk pages -- place them into a page with a relevant name.
If the problem is that you do not know how to use a Mediawiki, then please read the help pages.

I have taken the page you wrote in the Talk page for the Complex Analysis category and placed it here so you don't lose it.

Please also make an attempt to learn the house style. The below is not it. --prime mover (talk) 20:47, 11 March 2014 (UTC)

Rouche's theorem for analytic functions.
If $f(z)$ and $g(z)$ be are analytic (holomorphic) functions inside and on a simple closed curve $C$ and if $|g(z)|<|f(z)|$ for $z\in C$, then $f(z)+g(z)$ and $f(z)$ have the same number of zeros inside $C$.

Proof.

Because $|f(z)|>|g(z)|\geq0$ on $C$ it follows that $|f(z)|\neq0$ and $|f(z)+g(z)|\neq0$ also. Let $N_1$ and $N_2$ be number of zeros of $f(z)$ and $f(z)+g(z)$, respectively, inside $C$. By argument's principle $\displaystyle N_1=\frac{1}{2\pi}\Delta_Carg[f(z)]$ and $\displaystyle N_2=\frac{1}{2\pi}\Delta_Carg[f(z)+g(z)]$ so \begin{eqnarray*} N_2 &=& \frac{1}{2\pi}\Delta_Carg[f(z)+g(z)] \\ &=& \frac{1}{2\pi}\Delta_Carg[f(z)][1+\frac{g}{f}(z)] \\ &=& \frac{1}{2\pi}\Delta_Carg[f(z)]+\frac{1}{2\pi}\Delta_Carg[1+\frac{g}{f}(z)]\\ &=& N_1+\frac{1}{2\pi}\Delta_Carg[1+\frac{g}{f}(z)] \end{eqnarray*} Let $\displaystyle\omega=1+\frac{g}{f}(z)$ is a point in range of $\displaystyle1+\frac{g}{f}(z)$ that is on it's graph. From assumption $|g(z)|<|f(z)|$ we have $$|\omega-1|=\Big|\frac{g}{f}(z)\Big|<1$$ so $\omega$ must be inside the circle $|\omega-1|<1$ for $z\in C$, that shows $\omega$ doesn't meet $0$ then $\displaystyle\Delta_Carg[w]=\Delta_Carg[1+\frac{g}{f}(z)]=0$ and we conclude $N_2=N_1$.