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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37 views

Why this map is a mobius transformation

Question: Let $D_2=\bar D(2,1)$ and $D_{-2}=\bar D(-2,1)$ be the closed disks of radius $1$ centered at $z=2$ and $z=-2$ in the complex plane, respectively. Set $X= \mathbb C-\{D_2 \cup D_{-2} \}$, ...
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12 views

Computing the laplacian Green function by Fourier transform and analytic continuation

I know that the Green function for the laplacian operator in $d$ space dimensions $$\Delta \equiv \sum_{i=1}^d \frac{\partial^2}{\partial x^2_i}$$ is given by $$ \Delta^{-1}(x-x')=\begin{cases} ...
3
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0answers
42 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
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0answers
23 views

How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
4
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1answer
48 views

Are bounded analytic functions on the unit disk continuous on the unit circle?

Let $f(z)$ be holomorphic on the open disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Moreover, let $f$ be bounded on the boundary of $\mathbb{D}$, i.e. $$ \sup_{\varphi \in [0,2\pi]} ...
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65 views

Series $1$, $1$, $\frac12$, $\frac12$, $\frac13$, $\frac13$, etc.

Is there any way to define an analytical function in a region that's contained by 0 and 1 that will correspond with the following series: $1$, $1$, $\frac12$, $\frac12$, $\frac13$, $\frac13$, ...
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23 views

How to apply Cauchy-Kowalevsky Theorem.

The Cauchy-Kowalevsky theorem is stated in my notes as: For the Cauchy problem: $$ \begin{cases} u_{y}=F(x,y,u,u_{x}) \\ u(x,0)=h(x) \end{cases} $$ If $h$ is analytic in a neighborhood of ...
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3answers
57 views

Show that f must be constant on C

This is a problem that I have been encountered after reading about analytic functions in complex analysis. Suppose $f(z) = f(x + iy)$ is analytic on $\mathbb{C}$. Let $u= \Re ~f$ and $v = \Im ~f$. ...
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65 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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1answer
51 views

Not able to understand a paragraph in John Conway's Complex analysis book.

On page 97 under the heading "Counting zeroes; the open mapping theorem" there is a second paragraph which goes like this: In section 3 it was shown that if an analytic function $f$ had a zero ...
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1answer
53 views

Prove that $\,\displaystyle f(z) = \sum_{n\ge1}\frac{z^n}{n^2}$ 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 $D\big(\frac23\big)$. There is the ...
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2answers
40 views

smooth vs. analytic in the definition of almost-complex manifolds

Let $A_{\infty}\hspace{-0.03 in}$ be a maximal $C^{\infty}\hspace{-0.02 in}$ atlas on $M\hspace{-0.03 in}$, and with that smooth structure on $M$, suppose $\: j : TM\to TM\:$ is a smooth function ...
0
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0answers
59 views

Let $f$ be an entire function $\mathbb C$, and $g(z) = \overline { f(\overline z)}$. Which of the following statements are valid? [duplicate]

Let $f$ be an entire function, $\mathbb{C}$, and $g(z) = \overline{f(\overline{z})}$. Which of the following statements are valid? Let $f(z) \in \mathbb{R}, \forall z \in \mathbb{R}$, then $f = g$. ...
0
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0answers
71 views

Measure of inverse image of points by an analytic mapping

How can one prove the following statement: For any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of $M$ ...
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0answers
20 views

Schwarz Reflection Principle for the four quadrants in the plane and for two intersecting circles,

I'm looking at an old exam problem that shows a picture of what the function f does to the plane. On the upper right quadrant, there is a + sign, which indicates that f maps this quadrant one-to-one ...
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0answers
31 views

Understanding the Schwarz Reflection Principle,

The way I understand the principle is that we look at F, piecewise-defined as: F= $$ z \mapsto f(z), \ z \in \Omega^+$$ $$ z \mapsto \overline{f(\bar z)}, \ z \in \Omega^-$$ Here $\Omega^+$ and ...
0
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1answer
43 views

Show that the solution of this differential equation is analytic

Let $\alpha,\beta,a,b$ be real constants. Show that the differential equation given by: $y''= ay' + by \\ y(0)=\alpha\\ y'(0)=\beta$ has an unique solution and this solution is analytic in ...
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0answers
33 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
0
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0answers
27 views

Proof that real power series is real analytic

I'm wondering if the following argument is correct. The proof in the book is longer and I don't understand it. Theorem. Suppose $f(x) = \sum_{n=0}^\infty a_n x^n$, where the series converges for $-R ...
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0answers
18 views

Analytic structures on $S^1$|

I am currently studying Haefliger's paper "Homotopy and Integrablity". During the last chapter, he applies his theory of $\Gamma$-structures to analytic codimension $1$ foliations. Throughout the ...
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2answers
58 views

A question related to Montel's Theorem

Given $c>0$, there exist $\varepsilon > 0,$ such that, whenever $\{a_n\} \subset \mathbb C$ and $\sum_{n=1}^{\infty}\lvert a_n\rvert \le c\,$ implies that $$ \sup_{\frac{1}{2}\leq x\leq1}\left|1 ...
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2answers
35 views

What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$?

let $f:D=\{z\in \mathbb C:|z|<1\} \to \overline D$ with $f(0)=0$ be a holomorphic function. What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$? My try:By cauchy integral formula : ...
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1answer
29 views

Analyticity of log f(z)

In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is ...
2
votes
0answers
21 views

Application of Bernstein's theorem

There is a theorem due to Bernstein related to analytic functions : If $f : ]0,1[ \to \mathbb{R}$ is an absolutely monotonous function (that is a $\mathcal{C}^\infty$ function such that for ...
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0answers
17 views

Question about inverse of CDF being a real analytic function

Let F: [0,a] -> [0,1] be a continuous, strictly increasing CDF. Assume also F admists a continuous, positive pdf f. Now define the inverse function h(x) as F(h(x))=x. Is h real analytic? If not, what ...
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3answers
76 views

Analyticity of $\dfrac{1}{z}$ vs. $\dfrac{1}{z^2}$

I am learning complex analysis on my own. I am familiar with the theorems, and I am able to compute by hand and get correct results. But there is something that escapes me. What is the criteria for ...
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1answer
14 views

A Question from complex variable [closed]

Show that an analytic function with constant modulus is itself a constant
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63 views

Analytic Functions, Cauchys Integral Formula

Let $f: \mathbb D \to \mathbb D$ be analytic or holomorphic with $f(0)=\frac{1}{2}$ and $f(\frac{1}{2}) = 0$ where $\mathbb {D} = \{ z: |z| \leq 1\}$. Then find $|f^{'}(0)|$ and ...
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1answer
45 views

Find the residue of the function $g(z)=f(z^2)$ at a given point.

Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...
0
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1answer
61 views

Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
0
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2answers
59 views

Why does the ring of entire functions have no zero divisors?

Why does the ring of entire functions have no zero divisors, while the ring of infinitely differentiable functions on the real line does?
0
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1answer
55 views

Rouche Theorem and applications

Exercise 3. Let $f$ be analytic in $\overline{B}(0; R)$ with $f(0)=0$, $f'(0) \neq 0$ and $f(z) \neq 0$ for $0<|z| \leq R$. Put $\rho=\min\{|f(z)|:|z|=R\}>0$. Define $g: B(0; \rho) ...
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0answers
12 views

About smooth function whose reciprocal is also smooth

I know that for exponential functions $e^{at}$, or functions like $1/t^m$, t>0, both they and their reciprocals are smooth. Could you please give me more classes of smooth functions, or Analytic class ...
1
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1answer
50 views

How to find some $C^\infty$ functions that do not satisfy the uniqueness theorem for analytic functions

The uniqueness theorem for analytic functions states that suppose two series $\sum_{n=0}^\infty s_nx^n$ and$\sum_{n=0}^\infty t_nx^n$ converges in the interval $(-R,R)$. If the set of $x$ that ...
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0answers
38 views

An entire function is zero

Suppose $f(z)$ is an entire function that have zero on positive integers. Does it follow $f$ is identically zero? This seems like an application of Liouville theorem. But I cant come with a function ...
4
votes
2answers
42 views

$f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?

$f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$,but this ...
1
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1answer
81 views

Prove that there is no function $f$ that is analytic. [duplicate]

Prove that there is no function $f$ that is analytic in $\mathbb{C}\setminus\{0\}$ and satisfies $$|f(z)|\geq\frac{1}{\sqrt{|z|}},\quad \operatorname{for all}\quad z\in\mathbb{C}\setminus\{0\}$$ I am ...
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2answers
46 views

How to show real analyticity without extending to complex plane

Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$ For example, $$f(x)=(1+x^2)^{-1}.$$ Using complex analysis, we can easily show $f$ is real analytic. Is there an easy, general method ...
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1answer
30 views

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. ...
5
votes
2answers
82 views

Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: ...
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0answers
19 views

Prove that an entire function of exponential type is of order at most $1$.

By Entire functions theory, the order (at infinity) of an entire function $f(z)$ is defined using the limit superior as: $$\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln\Vert f \Vert_{\infty, B_r} ...
0
votes
2answers
47 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
0
votes
1answer
42 views

An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I ...
0
votes
1answer
89 views

Are the integrals of the following function path independent in the following domain?

Are the integrals of the function: $$f(z)=\frac{1}{z+1}+\frac{1}{(z+1)^2}+e^{\frac{1}{z}}$$ path independent in the following domain: $$D= \{Re z >0\}\setminus\{1\}$$ My thoughts on the ...
0
votes
1answer
34 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
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1answer
37 views

Bound on the sup norm for derivatives of a particular $C^\infty$ function

I'm reading textbook "A Primer of Real Analytic Functions" and on page 86 the following "obvious" claim is made: Let $|| \cdot ||$ be the sup norm on $[0, 2 \pi]$ and define function $f$ to be ...
0
votes
1answer
25 views

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. Then, (a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle; (b) the set {z : ...
2
votes
1answer
34 views

Prove $f$ analytic on $D(z_0;R)\setminus\{z_0\}$ implies $\exists M, f(D(z_0;r)\setminus\{z_0\})\supset\{z\in\mathbb{C}:|z|>M\}$

Suppose $f$ is analytic on $D(z_0;R)\setminus\{z_0\}$, and $z_0$ is a pole of $f$. Prove that for any $r\in(0,R)$, there is $M\in(0,\infty)$ such that ...
0
votes
1answer
52 views

Analytic continuation of function differentiable on real line to complex plane

If $f(z)=g(z)$ on $(0, \infty)$ and f(z) is holomorphic on an open set $U \subset \mathbf{C}$ with $(0, \infty) \subset U$, but we do not have any information about where $g(z)$ is holomorphic, can we ...
0
votes
1answer
37 views

derivatives of non-analytic smooth functions

I would like to know how to calculate the derivative of a non-analytic smooth function? Suppose $f:\mathbb R\rightarrow \mathbb R$ is in $\mathcal C^\infty\backslash \mathcal C^\omega$ and in ...