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

Does this property persists for the derivatives $f^{(k)}, k=1,2,..$

Let $f$ be a real non polynomial analytic function. Suppose that the function $f$ assumes arbitrarily large and arbitrarily small values, i.e., for all $K>0$, there are $a,b$ with $f(a)<−K$ and ...
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0answers
26 views

Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
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2answers
119 views

Complex Analysis Question About Analytic Functions

I have some questions about knowing where and where not functions are analytic. Here's a function, f(z)= $\frac{Log(z+4)}{z^2+i}$ -I know that this function is not defined for ...
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1answer
78 views

Analyticity of a real function on $[0,\infty)$

I'm struggling to understand the difference of the analyticity of a real and a complex functions. Consider the following real valued function which is a minimal example of a somewhat more involved ...
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1answer
33 views

Analyticity of Products

Assume we have two functions $f,g:\Omega\rightarrow\mathbb{C}$ that are analytic and a third function $h:\Omega\rightarrow\mathbb{C}$ with $f=g\cdot h$. Can one now show that $h$ is analytic as well? ...
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111 views

Is discontinuity along a line equivalent to branch cut?

Suppose I claim the analytic function $f(z)$ has a branch cut along the positive real line, how would one go on to prove this? Is it sufficient to prove that $f(z)$ is discontinuous across this line? ...
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0answers
103 views

Schwarz Reflection Principle for function complex on the real line

The Schwarz reflection principle is usually proved for function real on the real (or a subset of) line. I wonder if the same principle/theorem works for general analytic functions on the real line? ...
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3answers
913 views

Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...
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0answers
108 views

Applications of identity theorem to physics

Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the entire complex plane. So, just by knowing how the function behaves at a teenie-weenie open disc ...
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3answers
437 views

Where is $\log(z+z^{-1} -2)$ analytic?

I need some help in determining where $\log(z+z^{-1} -2)$ is analytic, where $z$ is a complex number and $\log(z)=\ln|z|+\arg(z+2k\pi),k\in\mathbb{Z}$. Thank you in advanced.
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1answer
75 views

Show that an analytic function is a limit of entire functions.

Let $V$ be an open subset of $\mathbb{C}$ and $B:=B(a,r)$ a ball whose closure is in $V$. Denote by $E$ the space of all complex-valued functions holomorphic on $B$ and continuous on its ...
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51 views

Is the case where the zeros of $f$ or $g$ are isolated possible? [closed]

Assume that $f,g:\mathbb{C}→\mathbb{R}$. Let us consider the following equation in $\mathbb{C}$ $$f(s)g(s)=0$$ My question is: What are the cases where the zeros of $f$ or $g$ are isolated?
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1answer
32 views

About a reference collecting the main properties of the modulus and the argument of $f$

Let $f$ be an analytic function in the whole complex plane. We can write $f$ in its polar form: $$f(z)=ρ(z)exp(iθ(z))$$ My question is about a reference collecting the main properties of the modulus ...
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1answer
93 views

Are the algebraic functions dense in the space of analytic functions?

I had a quick google, and couldn't ascertain a answer to the question 'Are the algebraic functions dense in the space of analytic functions over the interval [0,1]?' This is functions over one ...
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1answer
51 views

Dealing with partial derivatives in a function space

Please read the following details below. Question: I want to show now that if $r>s>0,f \in F_s (\Omega), $ and $u \in F_r (\Omega)$, then for any $i$, $$f \frac{\partial u}{\partial z_i} ...
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1answer
86 views

What does it mean to say that a function is valued in the space of analytic functions?

I am reading some paper and I encountered this statement: ... the coefficients $a_{p,\beta}(t,x)$ [are] of class $C^m$ in $t$, valued in the space of analytic functions of $x$, in a neighborhood ...
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2answers
59 views

What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$?

Let $s$ be complex and $\zeta(s)$ the Riemann Zeta function. What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$ ? I want a series expansion such that ...
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1answer
51 views

Analyticity of C*-algebra valued functions

Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally ...
3
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2answers
329 views

Proving that a function has a removable singularity at infinity

I'm having trouble with the following exercise from Ahlfors' text (not homework) "If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1} \Re f(z) \to0$ as $z \to \infty$, show that ...
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0answers
68 views

Infinite product of an analytic function on the right half plane

I want to know whether $G_t=\prod_{i=1}^\infty G(s)^i$ is defined and analytic on the right half plane for an analytic $G(s)$ on the right half plane. If so, is $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
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2answers
56 views

question involving analycity of $f=u+iv$

Let $f=u+iv:\mathbb C\to\mathbb C$ be analytic. Then is it true that $\dfrac{\delta^2 v}{\delta x^2}+\dfrac{\delta^2 v}{\delta y^2}=0?$
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1answer
71 views

Correct assertions about the real and imaginary parts of an analytic function

Let $f(z)=u+iv$ is analytic function, and $ u,v\colon\mathbb R^2 \to \mathbb R $ be such that $u(x,y)=Ref(z) $and $v(x,y)=Imf(z)$. Which of the following are correct? $u_{xx}+u_{yy}=0$ ...
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1answer
121 views

Trying to find more information about “Darboux's method/theorem” on coefficients of an analytic function

My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down ...
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1answer
81 views

Analyticity of Laplace transform

Let $f(x)$ be a bilateral Laplace transform of a measure $\mu$: $$ f(x)=\int_{-\infty}^{+\infty} e^{-xt} d\mu(y). $$ Suppose that $f(x)$ converges absolutely in $(a,b)$, and $(a,b)$ do not contain the ...
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395 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
4
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1answer
82 views

Analytic extension for a a function defined in $\mathbb{N}$

I would like to know if it is possible to extend analytically any function of the type $f:\mathbb{N} \to \mathbb{C}$ to all complex plane. If it isn't possible, what should I assume to do so? If ...
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2answers
312 views

Deduce that $f$ is constant in the following cases

We want to show that the function $f$, holomorphic on some domain $D$, is constant in the following cases: $z \mapsto \overline{f(z)}$ is holomorphic $z \mapsto f(\overline{z})$ is holomorphic ...
2
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3answers
226 views

Proof about Holomorphic functions in the unit disc

We want to prove the following: If $f$ is a holomorphic function on the unit disc $\mathbb{D}$ s.t. $f(z) \neq 0$ for $z \in \mathbb{D}$, then there is a holomorphic function $g$ on $\mathbb{D}$ ...
3
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1answer
66 views

Largest domain on which $z^{i}$ is analytic.

Can anyone help me with this question: What is the largest domain $D$ on which the function $f(z)=z^{i}$ is analytic?
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1answer
356 views

Complex differentiable but not analytic on circle of convergence

I'm trying to get a better handle on behavior of complex power series on the boundary of their maximal disk of convergence. I'm reading Bak-Newman's Complex Analysis, Chapter 18.1. A regular point ...
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4answers
377 views

what is the relation of smooth compact supported funtions and real analytic function?

What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
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1answer
68 views

Does this proof contain a contradiction?

I have made the following proof and I am asking if there is anything wrong in my steps: Let $f:\mathbb{R}\to\mathbb{R}$ be a real analytic function with infinitely many zeros. Let $a\neq 0$ be a real ...
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1answer
65 views

Analyticity implying Constant

Question: $f(z)$ is analytic in $C$ and $Im(f(z))\leq 0$. I want to show that $f(z)$ is a constant. Approach: I know that if $f$ is analytic on a closed curve then the line integral along that curve ...
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2answers
233 views

Using Montel's Theorem to show locally uniform convergence of analytic functions

Let $f_n :U \to \mathbb{C}$ be a sequence of analytic functions, where $U$ is open and connected. Suppose there exists a point $z_0 \in U$ such that for all $k \geq 0$ the sequence $f_n^{(k)}(z_0)$ ...
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1answer
251 views

Showing a sequence of analytic functions converges locally uniformly

Let $f_n :U \to \mathbb{C}$ be a sequence of analytic functions on an open and connected set $U$. Suppose that the sequence is locally bounded and that for the set $$D:= \{z \in U : f_n(z) \, \, ...
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123 views

Does $f'$ analytic imply $f$ analytic?

If $f'$ is known to be analytic, does it mean that $f$ is analytic as well? I've tried to expand $f$ and then to replace the tail of it by the expansion of $f'$, yet the factorials don't add up. I ...
1
vote
1answer
89 views

Find all the equivalence classes of $ℜ$

Let $f:ℝ→ℝ$ be a real analytic function. Assume that $f$ has infinitely may zeros. We define the equivalence relation $ℜ$ on $G(f)$ via $(x_1,f(x_1))ℜ(x_2,f(x_2))$ if and only if $f(x_1)=f(x_2)$. Here ...
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1answer
42 views

Can we deduce that $\lim_{x\to+\infty}f(x)=\pm\infty$ or $\lim_{x\to-\infty}f(x)=\pm\infty$?

Let $f:ℝ→ℝ$ be rael analtic function. Asume that $f$ is of finite order $1$ (An entire function is said to be of finite order if there exist numbers $a,r>0$ such that $$|f(x)|≤exp(|x|^{a})$$ for ...
0
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1answer
62 views

Find conditions on the function $f$ such that the fiber $f^{-1}(a)$ has a finite number of elements

Let $f:ℝ→ℝ$ be a real analytic function. We know that for any real number $a$, the fiber $f^{-1}(a)$ is a discrete set unless $f = a$. My question is: Find conditions on the function $f$ such that the ...
0
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1answer
124 views

Show that the fiber $f^{-1}(a)$ is finite if $a∈ℝ,a≠0$

Let $f:ℝ→ℝ$ be a real analytic function. If $f$ has infinitely many zeros, then we know that the fiber $f^{-1}(0)$ is an infinite discrete and countable set. Let $a∈ℝ,a≠0$, we know also that the fiber ...
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1answer
64 views

The fiber $f^{-1}(a)$ is a discrete (and countable) set

Let $f:ℝ→ℝ$ be a real analytic function. Then my question is: Show that for a real number $a$, the fiber $f^{-1}(a)$ is a discrete (and countable) set unless $f = a$.
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50 views

Finding Analytic Functions that Satisfy Certain Conditions

How would one go about finding: All analytic f such that $|f''(x)|>|e^{7z}|$? All analytic functions such that $f(z) = z + f(z^2)$. All linear functions that map from the punctured unit disk to ...
2
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1answer
206 views

$f, g$ are analytic at $z_0$. Then $f/g$ is analytic at $z_0$ iff $g(z_0) ≠ 0.$

How to show $f, g$ are analytic at $z_0$. Then $f/g$ is analytic at $z_0$ iff $g(z_0) ≠ 0.$ using $f, g$ are differentiable at $z_0$. Then $f/g$ is differentiable at $z_0$ iff $g(z_0) \neq 0.$ ...
3
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1answer
103 views

When does a function differentiable in a domain become constant

In the following result $f$ is differentiable on a domain $D$ & i) $\Re f$ is constant or ii) $\Im f$ is constant or iii) $\arg f$ is constant or iv) $|f|$ is constant on $D\implies f$ is ...
2
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2answers
178 views

Complex analytic function on a line

a) Let $D$ be a domain whose boundary $C$ contains a straight-line segment $L$. Let $f(z)$ be analytic in $D$ and continous on $L$. Assume also that $\Im(f) = v(x,y)$ vanishes on $L$. Prove that ...
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2answers
170 views

Check for analyticity of a complex function

Prove that $f(z)=|z|^4$ is differentiable but not analytic at $z=0$ My Attempt : $|z|=\sqrt{x^2+y^2} so |z|^4=(x^2+y^2)^2$ Now, we see that at $z=0$ all the four partial derivatives are equal to ...
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1answer
91 views

Prove there exists $f \in C^{\infty}$, bounded derivatives,

I know the title isn't very comprehensible, but I don't know how to improve it. Here is a problem which I don't know how to solve. Let $n \in \mathbb{N}, \ \ \alpha \in \mathbb{R}, \ \ \varepsilon ...
5
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0answers
79 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
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0answers
22 views

Analytic random function

Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function. I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic. What are the minimal conditions needed? ...
2
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2answers
304 views

Analytic functions of a real variable which do not extend to an open complex neighborhood

Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?