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Official GRE Super Power Pack (T ##HOT##

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Official GRE Super Power Pack (T

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read full description. Official GRE Super Power Pack 2/E Unlimitedbooks Download ePub Power.Q:

Prove that $f(P)=f(Q)$ if $P \in \mathcal{O}(V,0), Q \in \mathcal{O}(V,\alpha)$, $\alpha \in \mathbb{R}$

Consider the space $V=\mathbb{R}^2$, and two open sets $U \in \mathbb{R}^2, O=U \cap \mathbb{C}$, so $f:U \to V$ is a homeomorphism, $f^{ -1}:V \to U$ exists and is smooth, that is $f(p)=p$ for $p \in U$. Prove that $f(P)=f(Q)$ if $P \in \mathcal{O}(V,0), Q \in \mathcal{O}(V,\alpha)$, $\alpha \in \mathbb{R}$

$\mathcal{O}(V,x)$ is the set of germs of holomorphic maps $f:U \to V$ at $x \in V$.
My solution of the problem: since $f$ is a homeomorphism in its domain and range, for every $Q \in \mathcal{O}(V,\alpha)$ is uniquely defined $P \in \mathcal{O}(V,0)$ such that $f(P)=Q$, so let $f(P)=Q$, for all $Q \in \mathcal{O}(V,\alpha)$ it follows that $P \in \mathcal{O}(V,0)$, then since $f$ is homeomorphism, for

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Official GRE Super Power Pack (T ##HOT##

Official GRE Super Power Pack (T ##HOT##

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