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 ...

learn more… | top users | synonyms

3
votes
3answers
369 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.
2
votes
1answer
65 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 ...
0
votes
1answer
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?
0
votes
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 ...
0
votes
1answer
81 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 ...
1
vote
1answer
44 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} ...
0
votes
1answer
84 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 ...
0
votes
0answers
64 views

Determine whether this class of holomorphic functions is starlike

According to Singh and Singh [1], the class of holomorphic functions $f$ in the unit disk such that $f(0)=f'(0)-1=0$ and $\mathrm{Re}\{f'(z)+zf''(z)\}>0$ is a subclass of starlike function. My ...
0
votes
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 ...
5
votes
1answer
48 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
votes
2answers
257 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 ...
1
vote
0answers
65 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 $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
4
votes
2answers
55 views

question involving analycity of $f=u+iv$

Let $f=u+iv:\mathbb C\to\mathbb C$ be analytic. Then is it true that $\dfrac{\delta^2 v}{\delta x^2}+\dfrac{\delta^2 v}{\delta y^2}=0?$
1
vote
1answer
68 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? $u_{xx}+u_{yy}=0$ ...
1
vote
1answer
97 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 ...
3
votes
1answer
75 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 ...
8
votes
3answers
309 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
2
votes
1answer
77 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 ...
1
vote
2answers
292 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
votes
3answers
189 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
votes
1answer
58 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?
6
votes
1answer
322 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 ...
4
votes
4answers
324 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 ...
1
vote
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
votes
1answer
62 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
178 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
votes
1answer
197 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) \, \, ...
10
votes
2answers
122 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
82 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 ...
0
votes
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
votes
1answer
58 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
votes
1answer
117 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 ...
1
vote
1answer
61 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 $f = a$.
0
votes
0answers
46 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
186 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
votes
1answer
94 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
157 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 ...
1
vote
2answers
135 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
88 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
73 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 ...
1
vote
0answers
20 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
votes
2answers
220 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
votes
1answer
248 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 ...
1
vote
1answer
59 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
116 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
92 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
vote
1answer
87 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
73 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 ...
1
vote
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 $f$.
4
votes
1answer
271 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 ...