# 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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### Theorem 1.1 Part 2 Lang's Complex Analysis proof explanation

Page 294, Part 2 chapter 1 in Langs "Complex Analysis." Let $U^+$ be an open set in the upper half plane, and suppose $\delta U^+$ contains an interval $I$ of real numbers. Let $U^-$ be the ...
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### Prove that an analytic function bounded by a certain exponential is bounded. [duplicate]

Let $D=\lbrace z \in \mathbb{C} : Re(z)>0, \frac{-\pi}{2} < Im(z) < \frac{\pi}{2} \rbrace$ and $f$ a function analytic on $D$. Suppose $|f| \leq 1$ on the boundary of D and that there exists ...
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### If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$. [closed]

If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$. I could not find any confirmation regarding this solution of $a$ and $b$.
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### find all functions f, for which g is analytic

let f(z)=u+iv be analytic in open disk $D=\{z\in C s.t |Z|<1\}$. assume $u\neq v$ for all (x,y) in D. Find all such f, for which the funstion $g(z)=u^2+iv^2$ in analytic in D. My answer is: g is ...
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### Are two analytic functions equal if they are equal on the boundary of an open disk?

Let $D$ be the open disk in $\mathbb{C}$ with origin $0$ and radius $1$. Let $f,g: \overline{D} \to \mathbb{C}$ be continuous functions such that $f$ and $g$ are analytic on $D$ and such that $f=g$ ...
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### Why is the sum of two algebraic functions algebraic?

Let $U\subset\mathbb{C}^n$ be a domain. A holomorphic function $f:U\to \mathbb{C}$ is called $\textbf{algebraic}$ if there exists a polynomial $p(x,y)$ in the variables of $U\times \mathbb{C}$ such ...
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### does the scaling function of the coiflet wavelet have an analytical expression?

so I'm simply wondering if there exists an analytical expression for the scaling function associated with the Coiflet wavelet (specifically in the case S = 6).
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### Proof that $H(z)=\int_0^\infty h(z,t)\,dt$ is analytic?

Let $h(t,z)$ be a continuous complex-valued function defined for $0\leq t<\infty$ and $z\in D\subset\mathbb C$, where $D$ is a domain. Suppose that for each fixed $t$, $h(t,z)$ is an analytic ...
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### Conformal map from a 7-sided polyhedron to a square pyramid.

I have a right-angled square pyramid, $A$, whose height and base-length is $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
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### Conformal map from a quadrilateral to a triangle

I have a right-angled isosceles triangle, $A$, whose length and height are $l$. Now supposed I form a trapezium, $B$, by gluing a square, whose side-length is $l$ as well, to the vertical edge of $A$ (...
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### Complex Analysis 2.1.2 - 1 If g(w) and f(z) are analytic functions, show that g(f(z)) is also analytic. [closed]

Complex Analysis 2.1.2 - 1 If $g(w)$ and $f(z)$ are analytic functions, show that $g(f(z))$ is also analytic.
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### Doubt regarding proving analyticity of a power series

I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications" In the chapter of analytic continuation in basic concepts authors mention that ...
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### $x\mapsto \frac{1}{1+x^2}$ is analytic

Show that $f:\mathbb{R}\to\mathbb{R}, f(x)=\frac{1}{1+x^2}$ is analytic, i.e. that $\forall x_0 \in \mathbb{R}$, $f$ can be approximated in a neighbourhood of $x_0$ by a power series centred at $x_0$. ...
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### Real analyticity of harmonic functions

Let $u(x,y)$ be harmonic on a domain $D \in \mathbb C$, Can we say that $f(x)=u(x,0)$ is real analytic on $D \cap \mathbb R$ if it is non-empty? Clearly $f$ is infinitely differentiable on the domain,...
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### $f,g$ analytic on $I\subset \mathbb{R}$. If exist $a\in I$ such that $f=g$ and $f^{(n)}=g^{(n)}$then we have $f(x)=g(x)$ for every $x \in I$.

Problem Let $f,g$ analytic on an open interval $I\subset \mathbb{R}$. If exist some $a\in I$ such that $f(a)=g(a)$ and $f^{(n)}(a)=g^{(n)}(a)$ for all $n \in \mathbb{N}$ then we have $f(x)=g(x)$ for ...
Let $\mathcal{H}$ be a Hilbert space and let $f:(-1,1)^{n}/\{0\}\subset\mathbb{R} %EndExpansion ^{n}\rightarrow \mathcal{H}$ be an analytic function. If $f$ is bounded, does the limit $\underset{x\... 0answers 37 views ### A paradox on real vs complex analyticity We know two facts: -A complex function is complex analytic if and only if it is once complex- differentiable. -A real function is not necessarilty real analytic if it is once real-differentiable, ... 0answers 15 views ### PDE Analytic System Can someone give me some help with the following problem? I think it relates with Cauchy-Kovalevskaya Theorem, or maybe some technique from its proof. Let$u=(u_1, ..., u_N)$and consider the ... 0answers 18 views ### PDE System and Analytic Functions Please, could someone give me some help with this problem? I've struggling for days on it. Let$a, b_{i j}$be analytic functions defined in a neighborhood of$0$in$\mathbb{R}$and$\varphi ,\...
I am dealing with the following situation: I have a real analytic system of equations, $F(x,t)=0$, with $F: [0,1]^n \times [0, \infty) \rightarrow \mathbb{R}^n$. I know that only a single solution, ...