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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361 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
67 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 ...
2
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1answer
64 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 ...
5
votes
2answers
224 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 $f_n^{(k)}(z_0)$ ...
2
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1answer
228 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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2answers
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
85 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 ...
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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
123 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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0answers
48 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
votes
1answer
202 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
101 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
votes
2answers
173 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
155 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 ...
0
votes
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
votes
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
21 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
279 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?
2
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1answer
279 views

Show that some $C^\infty$ real function is analytic

Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
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1answer
65 views

Continuous dependence of zeros on a parameter

Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals. Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation $$f_\lambda(x)=0\,.$$ Assume its ...
2
votes
1answer
128 views

Let $ f:\Delta \mapsto \Delta $ be an analytic and bijective mapping.

Let $ f:\Delta \mapsto \Delta $ be an analytic and bijective mapping. My question is whether this implies $ f (z)=kz $ for some $ k \in\mathbb {C} $ such that $| k|=1 $. Here, $\Delta :=\{z\in \mathbb ...
2
votes
1answer
93 views

Prove that $f: (a,b)→ℂ$ cannot have infinitely many zeros in $(a,b)$

I have the following nonzero analytic function: $f:ℂ→ℂ$. We will consider only the restriction $f: (a,b)→ℂ$, $a,b∈ℝ$ and $a<b$. My question is: Prove that $f: (a,b)→ℂ$ cannot have infinitely many ...
1
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1answer
90 views

Can Cauchy theorem be applied to $\log{(z)}e^{ixz}$?

I'm reading about asymptotic analysis on the integral $I(x)=\int_0^1{\ln{t}e^{ixt}}dt$. The book tells me that I can use Cauchy theorem to deform the contour into a rectangular contour:0->iT, ...
4
votes
2answers
74 views

Is a function being analytic considered as a local property?

Sorry for being pedantic... I was just wondering if analyticity of a complex function considered as a local property? Apparently differentiability is considered as a local property. But analyticity ...
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1answer
105 views

Assume that the set of values where $f^{(k)}≠0$ is finite

Let $f:ℝ→ℝ$ be a real analytic function. Let $f^{(k)}$ be the $k$th derivative of $f$. Assume that the set of values where $f^{(k)}≠0$ is finite, then what we can say about the function $f$.
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1answer
315 views

Is the inverse of a real analytic function still analytic?

If $f:D\to D'$, with $D, D'$ open subsets of $\mathbb{C}$, is a complex analytic invertible function with non-zero derviative, it's easy to see that $f^{-1}:D'\to D$ is analytic too. Indeed complex ...
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1answer
56 views

Zeros of the analytic limit of complex rational function

For $n\in\mathbb{N}$ let $r_n,\ s_n$ be two polynomials of $O(n)$ degrees with real positive coefficients and set $f_n=r_n/s_n$. Suppose there exists $c>0$ such that $\bullet$ if $z\in\mathbb{C}$ ...
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1answer
57 views

Zeros of the analytic limit of complex polynomials

For $n\in\mathbb{N}$ let $p_n$ be a polynomial of degree $n$. Suppose there exists $c>0$ such that $\bullet$ if $z\in\mathbb{C}$ is a zero of a $p_n$, then $|z^2+c|\leq c$ (note that in particular ...
0
votes
1answer
54 views

Prove that the $k^{th}$ derivative of $f$ has necessarily infinitely many zeros

I have the following question: Let $f$ be a real entire function, i.e., $$f(x)=∑_{n=1}^{∞}a_{n}x^{n}$$ with infinitely many zeros. Prove that the $k^{th}$ derivative of $f$ has necessarily ...
2
votes
1answer
319 views

Zeros set of analytic functions over complex plane with several variables

I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane ...
3
votes
1answer
130 views

Bound for analytic function from unit disk into punctured unit disk

Suppose $f$ is analytic in the unit disk $D$ and satisfies $0<|f(z)|<1$. Show that $|f(z)|\leq|f(0)|^{\frac{1-|z|}{1+|z|}}$ for all $z\in D$. I tried to work with $\log|f|$. It seems that ...
1
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1answer
78 views

Behavior at infinity.

Classify the behavior at $\infty$ for $$f(z)=\frac{\sin z}{z^2},\,g(z)=\frac{1}{\sin z},\,h(z)=\exp\left(\tan\frac{1}{z}\right).$$ So I just considered $f(1/z),g(1/z),h(1/z)$ at $z=0$. For $f$ I ...
2
votes
1answer
294 views

Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$

What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$? Here's a plot ...
1
vote
2answers
290 views

Removable singularity and laurent series

How to show $z=\pm\pi$ is a removable singularity for $\frac1{\sin z}+\frac{2z}{z^2-\pi^2}$? I tried to compute the Laurent series, specifically the coefficients for $1/z,1/z^2,...$ since if we can ...
0
votes
2answers
236 views

Showing a bounded analytic function on strip is identically zero

Let $f$ be analytic and bounded on $\{x+iy\in\mathbb{C}:|y|<\frac{\pi}{2}\}$. Suppose $f(\ln n)=0$ for all $n\in\mathbb{N}$. Show that $f$ is identically 0. I tried to perform some transformations ...
2
votes
1answer
72 views

Evaluate $\int_C\frac{dw}{e^w-1}$ over some loop C contained in the annulus $0<|z|<2\pi$.

Evaluate $\int_C\frac{dw}{e^w-1}$ (counterclockwise) over some loop C contained in the annulus $0<|z|<2\pi$. Considering the coefficient of $1/z$ in the Laurent series for $\frac{1}{e^z-1}$ by ...
2
votes
1answer
258 views

Bounded imaginary part implies removable singularity at 0

Let $f$ be a holomorphic function on the punctured unit disk. If the imaginary part of $f$ is bounded, is it true that $f$ has a removable singularity at 0? I see that $|e^{-if}|=e^{Im\;f}$ so ...
0
votes
1answer
122 views

Does replacing analytic by continuity in open mapping theorem holds the result? and is it true for closed sets?

1) My first question depends on the open mapping theorem Which state that a non constant analytic function maps open sets onto open sets does the result holds if we use continuous function instead of ...
2
votes
1answer
252 views

Can someone please explain Morera's Theorem and the Schwarz Reflection Principle?

I have been reading Complex Analysis, Third Edition by Joseph Bak and Donald J. Newman and am stuck with chapter 7 which deals with the mentioned theorem. I have read through the chapter at least ...
1
vote
1answer
96 views

Analytic continuation of function given as integral

I have a function $I(D)$ defined by the following integral representation $$ I(D)=\int_0^\infty\mathrm{d}\alpha\,(1+2\alpha)^{-D/2} $$ which is clearly only sensible for $D>2$. The result of the ...
3
votes
2answers
187 views

If $f$ is analytic in a disk $|z|<R$ then so is $g(z)=\overline{f(\bar z)}$ in the disk

How to prove that if $f$ is analytic in a disk $|z|<R$ then $g(z)=\overline{f(\overline z)}$ is also analytic in the disk and also $f=g$ iff $f$ is real valued in $(-R,R)$
4
votes
1answer
120 views

If $f$ is analytic where $f$ is represented as $f=g.h$ where $g$ is analytic . From here can we conclude that $h$ is analytic?

If $f$ is analytic, where $f$ is represented as $f=g \cdot h,$ where $g$ is analytic. From here can we conclude that $h$ is analytic?
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votes
1answer
145 views

Complex has became so hard after the min\max modulus principle. Need some proofs and examples. [closed]

1) $f(z)$ being non constant and analytic in a domain $D$ if $f(z)$ continuous on $\overline{D}$ and $|f(z)|$ is constant on the boundary I need to prove that $f(z)$ must have a zero inside $D$! 2) ...
1
vote
1answer
285 views

Composition Taylor Series

Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name? Thanks.
3
votes
1answer
153 views

If$f(z)$ is analytic , then what about $f'(z)?? $

If$f(z)$ is analytic , then what about $f'(z)$? can we conclude that $f^{(k)}(z)$ is analytic for any k$\in $$ \mathbb{N} $
2
votes
1answer
652 views

Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.

I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
3
votes
1answer
162 views

Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
2
votes
3answers
204 views

Entire function dominated by another entire function is a constant multiple

These two questions I didn't even find the way to solve So please if you can help Suppose $f (z)$ is entire with $|f(z)| \le |\exp(z)|$ for every $z$ I want to prove that $f(z) = k\exp(z)$ for some ...