# 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.

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### Asymptotic expansion of non analytical function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function. Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders ...
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### Proving whether a complex function is analytical help!

I am currently struggling to prove whether a complex function is analytical. I understand that I must employ the Cauchy-Riemann relations to do this. However, the answer I get is one that I can't ...
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### Suppose that $f(z) = \frac{z}{1-z}$. Use Cauchy-Riemann Equations to determine the analyticity of $f(z)$. Then find the derivative of $f(z)$. [closed]

can anyone help me in solving this question? I have tried expressing $z=x+iy$ and multiplying the conjugate to numerator and denominator. However, the expansion and differentiation are too tedious. Is ...
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### Holomorphic is much stronger than real differentiable in a neighbourhood. What about analytic?

For real of 1 variable: Analytic $\implies$ smooth $\implies$ (real-)holomorphic $\implies$ differentiable (at a point $p$). here, real holomorphic at $p$ just means real differentiable in an open ...
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### If $f$ is analytic on an open and simply-connected subset of $\Bbb{C}$, and if $\cos{f}$ is constant, then must $f$ also be constant?

I need help with the following question: Suppose $f$ is analytic on the open and simply connected space $\Omega\subset\mathbb{C}$. Must $f$ be a constant function if $\cos{f}$ is constant? To me it ...
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### Domain of $I(z) = \int_0^\infty \frac{u^{z-1}}{1+u} du$, where z is complex and is it analytic?

Consider the function $I(z)$ defined through the integral $I(z) = \int_0^\infty \frac{u^{z-1}}{1+u} du$, where z is a complex parameter, what is the maximal choice for the domain of the function and ...
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### Is $f(r \cos \theta, r \sin \theta)=\theta$ real-analytic?

Let $U=\{(x,y) \in \Bbb R^2: y>0\}$ and $f:U \to \Bbb R$ defined by, $f(r \cos \theta, r \sin \theta)=\theta$. Is this function real-analytic? I know that polynomials in usual $x,y$ coordinates ...
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### extension problem with motivation

I am motivated by the Lorenz curve used in economics and statistics - a proper multivariate generalization will help researchers (Arnold, Taguchi, etc.) assess multidimensional risk and inequality (...
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### Existence of Analytic and non constant function

Can we construct a non constant analytic function $f(z)$ on open unit disc such that: $\frac{1}{\sqrt{n}} \lt |f(\frac{1}{n})| \lt \frac{2}{\sqrt{n}} \forall n \in \mathbb N$ My approach: I think ...
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### Cauchy-Riemann equations and required continuity of derivatives

So I just read, that for any analytic function, the Cauchy-Riemann equations will hold. However, the reverse, i.e. Cauchy-Riemann equations hold -> function is analytic, is supposedly only true if ...
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I am having problem with the following question. Have: $f(z)$ be analytic on an open set $\Omega$ with $\overline{\Delta(0, r)} \subset \Omega$. for zeros of $f$ in $\Delta(0, r)$ are $a_{1}, a_{2}, \... -1 votes 1 answer 60 views ### Can you always specify the value of a$C^{\infty}$or analytic function on an isolated set? [duplicate] Former math grad student, now a lawyer for the last$28$years. Just doing math for fun in my spare time. I was browsing questions here and my mind went on a tangent. The following questions ... 2 votes 1 answer 62 views ### Is$ z / (e^z - 1) $analytic in$ |z| < \pi $? I see that$ z / (e^z - 1) $has a singularity at$ z = 0 $. Is$ z / (e^z - 1) $analytic everywhere in$ |z| < \pi $? If yes, how is it determined to be analytic everywhere despite a singularity ... 0 votes 0 answers 50 views ### Question on the definition of analytic continuation along a path In my notes I have that$(g,E)$is an analytic continuation of$(f,D)$along a path$\gamma :[0,1]\to U$if there exists function elements$(f_i,D_i)$with$i\in \{ 0,\ldots ,n\}$and$0=t_0<t_1<...
I am having doubt regarding the power series of the function $g(x) = \frac{1}{1+x^2}$. Since the function $f(x) = {1+x^2}$ is analytic everywhere and by using the theorem that if $f(x)$ is analytic ...