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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28 views

Number theory about lehmers equations [on hold]

1) Find $n$ such that $\phi(n)$ divides $n-2$ 2) Find $n$ such that $\phi(n)$ divides $n+2$ 3) find $n$ such that $\phi(n)$ divides $2n\pm 2$ 4)find $n$ such that $\phi(n)$ divides $8n \pm 2$
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32 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
0
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1answer
20 views

Finding an analytic function

enter image description here I cannot find any such function. Also, why would a function that is analytic at 0 following these criteria not be analytic on (-2,0). Thanks in advance for your help.
4
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2answers
476 views
+50

Analytic continuation of a real function

I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique. Now i am wondering if this is ...
2
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1answer
32 views

Identity theorem for forms (on analytic manifolds)

It is well-known that for holomorphic functions the Identity theorem holds: if two holomorphic functions agree on an open subset, they agree everywhere (assuming the manifold connected). I would ...
15
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3answers
538 views

Can a function “grow too fast” to be real analytic?

Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for all real analytic functions $\: g : \mathbf{R} \to \mathbf{R} \:$, for all real numbers $x$, there exists ...
1
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1answer
40 views

Application of Schawarz lemma??

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. We have $|f(z)|\leq 1-|z|$ for all $z$ ...
3
votes
2answers
67 views

an analytic function being zero

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1\}$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. Please give only hints.
2
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0answers
34 views

$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
0
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1answer
30 views

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...
0
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1answer
59 views

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} ...
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0answers
23 views

Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to ...
0
votes
1answer
14 views

Domain of holomorphicity of $f(z)=\frac{1}{e^z+1}$

Domain of holomorphicity of $$f(z)=\frac{1}{e^z+1}$$ Is this simply everywhere, because there is no $\bar z$ dependence?
2
votes
3answers
63 views

$f(|z|)$ is not an analytic function

Let $f: [0,\infty)\rightarrow \mathbb{C}$ is a non constant function. Define $g:\mathbb{C}\rightarrow\mathbb{C}$ by $g(z)=f(|z|)$. Prove that $g(z)$ is not holomorphic. So, I need to find a point ...
3
votes
2answers
53 views

Powers of a function being analytic [duplicate]

Question is as follows : Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is continuous such that $f^3,f^4$ are analytic in $\mathbb{C}$ then prove that $f$ is analytic in $\mathbb{C}$.. Choose ...
1
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1answer
41 views

Entire function bounded sequence

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an entire function such that $|f(1/n)|\leq 1/n^{3/2}$ for all $n\in \mathbb{N}$ then show that $\{n^2f(1/n)\}$ is bounded. To show that $\{n^2f(1/n)\}$ is ...
2
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1answer
53 views

Power of a function is analytic [duplicate]

Question is : Show that if $f(z)$ is continuous function on a domain $D$ such that $f(z)^N$ is analytic on $D$ for some integer $N$ then $f(z)$ is analytic on $D$.. For some time i was wondering ...
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0answers
20 views

Function analytic in each variable does not imply jointly analytic

I have heard that a function $f: \mathbb R^2 \to \mathbb R$ can be analytic in each variable (i.e. $f(x,y_0) = \sum_{n=0}^{\infty} a_n x^n, \forall x \in \mathbb R$, and the same for $y$) without ...
5
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0answers
174 views

Laurent expansion of $\frac{1}{\sin z}$

Question is a fully solved exercise in Gamelin's complex analysis. Exercise : Consider the Laurent series expansion for $\frac{1}{\sin z}$ that converges on the circle $\{|z|=4\}$. Find the ...
0
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1answer
17 views

Is this function analytic: $F(x+iy)=\frac1{\pi}\int_{\mathbb R}\frac y{(x-t)^2+y^2}\,f(t)\,dt$.?

Let $f:\mathbb R\to\mathbb R$ such that $f(t)=0$ if $|t|\leq 1$ and $f(t)=|t|^\lambda$ if $|t|>1$. Here $\lambda<0$ is a constant. We consider the following function on $\mathbb ...
1
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2answers
64 views

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. [closed]

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. For what functions do we have equality? I'm not sure ...
1
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1answer
28 views

Analytic, non- constant complex function has finitely many zeros inside the disk $D(0, R)$ for all $R > 0$

I'm learning about complex analysis and need help with the following problem: Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and non-constant. Show that for every $R > 0$, the complex function ...
0
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0answers
10 views

Mobius transformations

Suppose f is a continuous function on the extended complex plane which is analytic except possibly at one point and maps lines and circles to lines and circles. Does it follow that f is necessarily a ...
0
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1answer
15 views

Interior Uniqueness: Does there exist an analytic function on a neighborhood of $z=0$ that satisfies the following?

I am faced with the following problem: Does there exist a function that is analytic on a neighborhood of $z=0$ and satisfies the following condition for every positive $n$: (a) ...
0
votes
0answers
25 views

Local Uniform Convergence and Composition

I've been sitting down can't quite tell if this is true or not, but I suspect that it should be. Edit: Suppose that $\Omega$ is a open, connected subset of $\mathbb{C}$, and suppose that $(f_n) ...
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28 views

How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
1
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0answers
35 views

function analytic in the entire complex plane is constant

Let the function $f$ be analytic in the entire complex plane and suppose that $\frac{f(z)}{z}\rightarrow 0$ as $|z|\rightarrow \infty$. Prove that $f$ is a constant. As $\frac{f(z)}{z}\rightarrow 0$ ...
0
votes
2answers
47 views

Why is this point analytic?

Suppose you had the function $$ p(x) = \frac{\sin(x)}{x} $$ I know, from other material online, that this point is analytic at the point $x=0$. However, my understanding was that a point of a function ...
0
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0answers
13 views

How is it possible to find all singular (non-analytic) points of a differential equation?

Suppose we had a second order differential equation of the form $$ y'' + p(x)y' + q(x)y = 0 $$ I know that a point $x=x_{0}$ is said to be ordinary if $p(x_{0})$ and $q(x_{0})$ can be expressed as an ...
1
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2answers
35 views

Laurent expansion

Question is to write Laurent expansion of $f(z)=\dfrac{1}{(z-2)(z-1)}$ in the annulus $1<|z|<2$ based at $z=0$ I am aware of the method of partial fractions and writing expansions for both ...
0
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1answer
13 views

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 ...
3
votes
1answer
88 views

Analytic function on unit disk has finitely many zeros

I am studying complex analysis from Theodore Gamelin's text and Exercise 1 of chapter IX.2 says that if $f$ is analytic inside the open unit disk and continuous on its boundary that satisfies $|f(z)| ...
1
vote
1answer
35 views

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 ...
0
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1answer
16 views

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 ...
2
votes
1answer
25 views

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 $x \in R$. ...
2
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0answers
46 views

A single analytic function that can approximate all others

The problem in it's entirety is this: Given some simply connected domain $U$ then show that there is a function $g\in\mathcal{O}(\mathbb{C})$ such that for any given $f$ there exists a sequence ...
1
vote
1answer
14 views

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 ...
0
votes
1answer
22 views

Let $f(z) = u(x,y) + iv(x,y)$ be analytic in $\Omega$, suppose that $v(x,y) = e^{-y}(y\cos x -x \sin x)$, find $f(z)$.

Question: a) Let $f(z) = u(x,y) + iv(x,y)$ be analytic in $\Omega$, suppose that $v(x,y) = e^{-y}(y\cos x -x \sin x)$. Find $f(z)$. b) Let $f(z), g(z)$ be analytic in an open, connected domain ...
1
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1answer
33 views

Bounded holomorphic function, limsup, boundary

Let $f \in \mathcal{O}(D)$, $\ D$ is a bounded region in $\mathbb{C}$, be such that $$\limsup _{D \ni z \to z_0} |f(z)| < \infty$$ for any $z_0 \in \partial D$. Prove that $f$ is bounded. So we ...
4
votes
1answer
53 views

Approximating polynomials over $\mathbb{C}$ with an entire function

Given a series of polynomials $p_{n}$ and a series of non-intersecting balls $B_{n} \subset \mathbb{C}$ show that there exists some function $f \in \mathcal{O}(\mathbb{C})$ such that $lim_{n ...
1
vote
2answers
45 views

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 $p_{n}(x)=\prod_{m=k}^{n}(1+a_{n}(x))$ ...
13
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158 views

Is this function nowhere analytic?

One usually sees $f(x):=\exp\frac{-1}{x^2}$ as an example of a $C^\infty$ function that is not analytic, having one point of non-analyticity (the point $0$). The Fabius function is a canonical ...
0
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0answers
40 views

If $g(x_a)=g(x_b)$ implies $f(x_a)=f(x_b)$ where $f$ and $g$ are analytic, does there exist a Holder function $h$ such that $f = h\circ g$?

Let $\mathbb{R}^p$ be the $p$-dimensional Euclidean space. Let $\ f:\mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}^m$ be analytic functions. Assume these two functions are such ...
3
votes
1answer
38 views

Analyticity of solutions to the heat equation

Let us look at solutions to the linear heat equation on $\mathbb{R}$: $$ u_t = u_{xx}.$$ Is it true that solutions to the equation with nice enough initial datum are analytic after a certain time $T ...
0
votes
0answers
33 views

Is the solution of a polynomial an analytic function on the polynomial parameters?

Be $\mu(z_1, \ldots, z_L)$ the only positive real solution to the equation \begin{equation} \sum_{l=1}^L z_l \mu^l = 1 \end{equation} With $z_1 = 1$, $z_l \geq 0 \forall l$. Clearly, varying the ...
2
votes
1answer
23 views

Prove: Show $\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$ for $f(z)=z^n$ and any $n \in N$

Prove that if $f:\mathbb{D} \to \mathbb{D}$ (where $\mathbb{D}$ is the unit disk) is given by $f(z)=z^2$, the for all $z \in \mathbb{D}$, we have $$\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$$ ...
2
votes
1answer
16 views

Suppose $\{f_n\}$ is a sequence of analytic in a region $D$

suppose $\{f_n\}$ is a sequence of analytic in a region $D$ such that $|f_n(z)|\leq M_n$, where $\sum_n M_n<\infty$, and $\lim_{z\to z_0}f_n(z)=L_n$. Show that if $p=p(z)\to\infty$ as $z\to z_0$, ...
1
vote
0answers
48 views

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges. Is $$F(z)=\int_0^\infty g(t)\sin(zt)dt$$ analytic? And if so, in what region? My attempt: ...
0
votes
0answers
25 views

Holomorphical extension to the Annulus

Let $D=\{z:1<|z|<2\}$ and $f$ is holomorphic on $D$. Suppose that f has a primitive $f_1$ on D and $f_1$ also has a primitive $f_2$, etc for every $n$ $f_n$ has a primitive $f_{n+1}$ in $D$. How ...
1
vote
0answers
66 views

Simply connected domains and complex logarithms

While studying Complex Analysis from my professor's notes I came across the following theorem. A demain $D$ in the complex plane is simply connected if and only if any analytic function $f(z)$ on ...