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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18 views

curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider ...
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38 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
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1answer
59 views

Application of Baire Category theorem

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to ...
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134 views

Conjecture on zeros of analytic function

I have a conjecture that I can´t prove nor disprove, any help on doing so will be very grateful. Let $f: \{z: |z|<2\} \to \mathbb C$ be a non constant analytic function such that if $|z|=1$ ...
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74 views

Identity Principle type question: Prove that $f=g$

While reading a complex analysis textbook the following assertion came up Since $f,g:D\equiv D(a,r) \to \mathbb{C}$ are analytic and injective functions such that $f(D)=g(D)$, $f(a)=g(a)$ and ...
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18 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
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42 views

Taylor expansion in terms of known Taylor series

Let $f(x)$ be a "nice" function with the following properties: $f$ is a real analytic, strictly increasing, odd and bounded function, i.e. $f(-x)=-f(x)$ and $-1<f(x)<1$. Further, let $f(x)>0, ...
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1answer
43 views

Is an analytic one-to-one function on the whole plane necessarily a polynomial? (Can it be disproved?)

I had to show what a one-to-one analytic function from the plane to itself could possibly be. So, I studied the behavior of such a function at infinity: Case 1: Such a function cannot have no ...
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1answer
48 views

Characterization of analytic functions

First, see this link on the alternative characterizations of analytic functions. I want to prove a version of 3) for complex-analytic functions. In particular: If $f$ is a complex-analytic function ...
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111 views

Is this a Morera´s Theorem Application?

Let $G \subset \mathbb C$ be a domain and $f: G \to \mathbb C$ a continuous function such that for any closed and rectifiable path $\gamma \subset G$, $$ \left| \oint_\gamma f(z)dz \right|\leq \left( ...
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58 views

What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty ...
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1answer
38 views

Analytic continuation of a certain Dirichlet series

Is there an elementary way to analytically continue $$f(s)=\sum_{n=1}^\infty \frac{(-1)^n}{(2n+1)^s}$$ to the entire complex plane? It is not hard to see (by grouping terms in pairs and using the ...
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1answer
27 views

How to prove that a Banach space of analytic functions containing $H^\infty$ except the origin is simply connected?

If $X$ is a Banach space of analytic functions on the unit disk $D$ which contains the space of analytic bounded functions on $D$, how can I prove that $X\setminus\{0\}$ is simply connected?
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1answer
75 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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13 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} ...
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45 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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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 ...
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1answer
55 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
58 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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78 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
53 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
54 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
42 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 ...
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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 $f = g$. ...
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72 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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23 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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32 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 ...
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1answer
45 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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42 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 ...
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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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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
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
30 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
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0answers
24 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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18 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
77 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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2answers
65 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
47 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 ...
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1answer
73 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
61 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?
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1answer
60 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 ...
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1answer
52 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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2answers
47 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 ...
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1answer
88 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
48 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
34 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. ...