Tagged Questions

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties ...

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The existence of the roots of an holomorphic function on an open connected domain

Let $U$ be an open connected domain and $D$ be an open disk such that the closure of $D$ is a subset of $U$. Suppose $f\in H(U)$, i.e., $f$ is holomorphic in $U$, and that $f$ is not constant. Show ...
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Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
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Evaluate $\int_{|z|=1}\frac{|dz|}{|z-a|^2}$ using Cauchy's Theorem.

Question: Evaluate $\int_{|z|=1}\frac{|dz|}{|z-a|^2}$ using Cauchy's Theorem. My attempt: So, Cauchy's theorem for derivatives tells us that if $f$ is holomorphic in an open set $\Omega$, and $D$ is ...
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Let the function $f$ be analytic in $C$, real valued on $R$, and $\Im f(z) > 0$ in the upper half-plane. Prove that $f'(x) > 0$ for $x \in R$.

Question: Let the function $f$ be analytic in the entire complex plane, real valued on the real axis, and of positive imaginary part in the upper half-plane. Prove that $f'(x) > 0$ for $x \in R$. ...
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Analytic function on the annulus $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ s.t. $C_1 \to C_1$, $C_4 \to C_2$?

Question: Does there exist an analytic function mapping $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ and taking $C_1 \to C_1$, $C_4 \to C_2$, where $C_r$ is the circle of radius $r$...
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Some proofs about the infinite products; the series and the analyticity

Definition A.1.4. The infinite product $\prod_{n=1}^{\infty}(1+a_{n}(x))$, where $x$ is a real or complex variable in a domain, is uniformly convergent if $p_{n}(x)=\prod_{m=k}^{n}(1+a_{n}(x))$ ...