Questions tagged [analyticity]

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 of functions over the complex numbers.

If $U$ is a subset of $\mathbb{C}$, $f$ is analytic at $x_0$ if there exists a series $$ \sum_{j=0}^\infty a_j (z-z_0)^j $$ that converges to $f$ at a neighbourhood of $z_0$. In complex analysis, analyticity is equivalent to homomorphy.

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Find the domain of analyticity of $f$

Find the domain in $\mathbb{C}$ where the function $f(z) = \sqrt{z^2 - 1}$ is analytic. So far I've tried to take the derivative of $f$ and got $f'(z) = \frac{z}{\sqrt{z^2 - 1}}$ which is not defined in $\mathbb{C}$ at $z = \pm1$ However that is not…
Pedro
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Suppose $f$ is analytic and $f(a) = f(b) = 0$. Show that $|f(z)| ≤ |{z − a \over 1 − z\bar{a}}| · |{z − b \over 1 − z\bar{b}}|$.

Suppose $f$ is analytic from $D(0, 1)$ to $D(0, 1)$ and $f(a) = f(b) = 0$ for two different numbers $a, b$ in $D(0, 1)$. Show that $\left\vert f(z) \right\vert ≤ \left\vert{z − a \over 1 − z\bar{a} }\right\vert \cdot \left\vert {z − b \over1 −…
Happy
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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 that if $|f|$ is constant on the boundry of $D$,…
Math1995
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Example of two analytic functions that differ at countably infinity many point

$\displaystyle f_1(x) = \frac{x^n-1}{x-1}$ and $f_2(x) = x^{n-1} + \cdots + 1$ have the same values except at $x=1$ (where $f_1$ fails to be analytic ). Is there an example of two analytic function that differ at infinitely many countable point?
jimjim
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Prove that $ f(z) = \sum\limits_{n\ge1}\frac{z^n}{n^2}$ is univalent in the disk $\,D\big(\frac23\big)$

I'm having some difficulty with this question: Prove that the function $\,\,\displaystyle f(z) = \sum_{n=1}^\infty\frac{z^n}{n^2}\,$ is univalent in the disk $D\left(\dfrac23\right)$. There is the following hint: $\dfrac{z^n-w^n}{z-w} =…
amirbd89
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A Question from complex variable

Show that an analytic function with constant modulus is itself a constant
varun
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Solving a logarithmic equation with variables on each side

Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. This really bothers me and there must be a way to…
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$e^{\mathrm{Re}\,z}$ not analytic in complex plane

In my textbook I found a text where it says that $e^z$ (z is a complex number) is analytic everywhere. But $e^x=e^{\mathrm{Re}\,z}$ is not. How can I prove that about $e^x$ and what is the difference?
Sijaan Hallak
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