- #1

mathmari

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MHB

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**Question 1:**

If $f\in O(\Delta (0,1,15))$ then does it hold that $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=2\pi i\text{Res}\left (\frac{f(z)}{(z-6+4i)^5}, 6-4i\right )+\int_{C(0,6)}\frac{f(z)}{(z-6+4i)^5}\, dz$$ Do we maybe use here Cauchy theorem and then we get $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=0$$

**Question 2:**

Is there a sequence of holomorphic polynomials $P_n(z), n=1,2,\ldots$ such that $$P_n(z)\rightarrow \frac{z(e^{6z}-1)(e^{4z}-1)}{\sin^2\left (\frac{z}{4}\right )\sin^2\left (\frac{2z}{5}\right )}$$ as $n\rightarrow \infty$, uniformly for $z\in \Delta (0,1,3)$ ?

since the convergence is uniform we get that $$\int \frac{z(e^{6z}-1)(e^{4z}-1)}{\sin^2\left (\frac{z}{4}\right )\sin^2\left (\frac{2z}{5}\right )}\, dz=\lim_{n\rightarrow \infty}\int P_n(z)$$ Do we have tocheck if this integral is equal to $0$ ?:unsure: