Mathematics Asked on December 7, 2020

Show that $lim_{Rtoinfty}$$oint_{|z|=R}$ $frac{P(0)}{zP(z)} =0$. We are given that $P(z)$ is a non-constant polynomial in $z$, and that $P(z) neq 0$.

Here is my attempt at the solution:

Since we have a circle centered at $0$ with radius R, the length of the circle we are integrating over is given by $2pi$$R$, so we can choose our L for the estimate to be $2pi$$R$. To find M, we can see that $frac{|P(0)|}{|z||P(z)|}$ $leq frac{|P(0)|}{|R||P(R)|}$. So we pick $M=frac{|P(0)|}{|R||P(R)|}$. Now by the ML estimate we have $lim_{Rtoinfty}$$oint_{|z|=R}$ $frac{P(0)}{zP(z)}$ $leq$ $lim_{Rtoinfty}$$2pi$$R$ $frac{|P(0)|}{|R||P(R)|}$ = $lim_{Rtoinfty}$$frac{|P(0)|}{P(R)}=0.$ As our integral is $leq 0$ we have that the integral itself must be equal to $0$. Is this my stopping point or is this more that I have to do here?

You are almost there. Note that $|int_{|z|=R}frac{P(0)}{zP(z)}, dz|leq int_{|z|=R}frac{|P(0)|}{|z||P(z)|}, |dz|$. Using $z=Re^{itheta}$ and parameterizing $0leq thetaleq 2pi$, we have $dz=Rie^{itheta}dtheta$ so $|dz|=Rdtheta$. Hence, an upper bound is $$ |P(0)|int_0^{2pi}frac{1}{R|P(Re^{itheta})|}R,dtheta. $$ The key is that as a function of $theta,$ we have some $theta_0$ such that $|P(Re^{itheta})|geq |P(Re^{itheta_0})|$. This holds since polynomials are continuous. Hence, you should find an upper bound that converges to zero.

Correct answer by zugzug on December 7, 2020

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