How Does the Argument Principle Relate Poles and Zeros of Analytic Functions?

[alyafey22]

$$\text{I will prove the Argument THM }$$

$$\text{If f(z) is a function that is analytic in a closed simple curved }\gamma$$
$$\text{ and has M Poles , and N Zeors inside }\gamma$$

$$\text{Then : }\frac{1}{2i\pi } \oint_{\gamma}\, \frac{f'(z)}{f(z)}\, dz \,=\, N-M$$

$$\text{Assume : }f(z) = \frac{P(z)}{Q(z)}$$
$$f'(z) = \frac{P'(z)Q(z) - P(z) Q'(z)}{(Q(z))^2}$$

$$\text{R.H }\frac{1}{2i\pi } \oint_{\gamma}\, \frac{f\frac{P'(z)Q(z) - P(z) Q'(z)}{(Q(z))^2}}{\frac{P(z)}{Q(z)}}\, dz$$

$$\frac{1}{2i\pi } \oint_{\gamma}\,\frac{P'(z)Q(z) - P(z) Q'(z)}{Q(z)P(z)}\, dz$$

$$\frac{1}{2i\pi } \oint_{\gamma}\,\frac{P'(z)}{P(z)}-\frac{ Q'(z)}{Q(z)}\, dz$$

$$\frac{1}{2i\pi } \oint_{\gamma}\,\frac{P'(z)}{P(z)}\,dz \, -\, \frac{1}{2i\pi}\oint_{\gamma}\,\frac{ Q'(z)}{Q(z)}\, dz$$

$$\frac{1}{2i\pi } \oint_{\gamma}\,\frac{P'(z)}{P(z)}\,dz$$

$$\text{Assume : }P(z) = (z-z_0)(z-z_1)...(z-z_N)$$

$$P'(z) = (z-z_1)...(z-z_N)+(z-z_0)...(z-z_N)+...+(z-z_0)...(z-z_{N-1})$$$$\frac{P'(z)}{P(z)} =\frac{ (z-z_1)...(z-z_N)+(z-z_0)...(z-z_N)+...+(z-z_0)...(z-z_{N-1})}{(z-z_0)(z-z_1)...(z-z_N)}$$

$$\frac{1}{2i\pi } \oint_{\gamma}\, \frac{ (z-z_1)...(z-z_N)+(z-z_0)...(z-z_N)+...+(z-z_0)...(z-z_{N-1})}{(z-z_0)(z-z_1)...(z-z_N)}\,dz$$

$$\frac{1}{2i\pi } \oint_{\gamma}\,\frac{1}{z-z_0}+\frac{1}{z-z_1}+...+\frac{1}{z-z_N} = N$$

$$\text{similarily we can show : }\frac{1}{2i\pi}\oint_{\gamma}\,\frac{ Q'(z)}{Q(z)}\, dz = M$$$$\text{Then : }\frac{1}{2i\pi } \oint_{\gamma}\, \frac{f'(z)}{f(z)}\, dz \,=\, \fbox{N-M} \text{ W.R.T}$$

Feel free to post any comments .

 

[jvicens]

Hi there,

Thank you for sharing your proof of the Argument THM. It seems like a solid and well-explained argument. I appreciate that you took the time to break down each step and provide explanations along the way.

One question I have is about the assumption that f(z) can be written as P(z)/Q(z). Is this a necessary assumption for the proof, or could it be generalized to any analytic function? Also, I'm not sure if you meant to include the R.H in the first line of your proof, as it doesn't seem to be used in the rest of the proof.

Overall, great job and thank you for contributing to the discussion on this topic. Keep up the good work!