Questions tagged [analytic-functions]
For questions about analytic functions, which are real or complex functions locally given by a convergent power series.
0
votes
1answer
45 views
Analytical Function at Real infinity
Let's consider a function $f(z)$ analytical everywhere except at infinity. It's zero at real infinity $\lim_{x\rightarrow \infty}f(x)=0$.
Is it true that $\lim_{x\rightarrow \infty}f(x+iy)=0$, where ...
0
votes
1answer
33 views
For a non-constant holomorphic function $f$ in a neighbourhood of $D(0,1)$ with $|f(z)|=1, \forall |z|=1$, $D(0,1)$ is contained in the image of $f$
Let $U\subseteq \mathbb C$ be an open connected set such that $\overline {D(0,1)} \subseteq U$ . Let $f: U \to \mathbb C$ be a non-constant , holomorphic function such that $|f(z)|=1, \forall |z|=1$. ...
1
vote
4answers
61 views
How do we know $\sum_{i=0}^n\frac{x^n}{n!} $ converges to $ e^x $ for all x? [duplicate]
$$\sum_{i=0}^n\frac{x^n}{n!} $$
I know the sum converges for all x but how do we know it converges to the expect value $e^x$.
This sum was derived as the Taylor series of $e^x$ around $0$. How do we ...
0
votes
0answers
30 views
What intrinsic property determines whether a function is analytic
Given we know the value of all order derivatives at a point $x_0$ for a given f(x).
As per my knowledge all the geometric properties like slope, curvature, convexity are functions of solely the ...
0
votes
1answer
23 views
Zeta function of the hypersurface of some homogeneous polynomial
Let $f(y)\in \Bbb Z_p[y_0,y_1,....,y_n]$ be a homogeneous polynomial. Let $N_s$ be the number of zeros of $f$ in $\Bbb P^n(F_{p^s})$. Here, $\Bbb P^n(F_{p^s})$ denotes the $n$-th projective space ...
1
vote
0answers
23 views
Criteria for an analytic function when the real and imaginary parts are $n$ dimensional
Say we have two functions
$$
f,g:\mathbb{R}^n\rightarrow\mathbb{R}
$$
and we define some complex function $\mathbb{R}^n\rightarrow\mathbb{C}$ as
$$
h(\vec x) = f(\vec x) + ig(\vec x)
$$
If $n=2$, ...
1
vote
1answer
24 views
Question regarding complex power series
Let us define a function $f$ on unit disc by-
$f(z)=\sum z^{2^n}\ \forall |z|<1$
Let $\theta=2\pi p/2^k$ where $p,k$ are positive integers and $0\le r<1$ so $f(re^{i\theta})$ exists.
Show that, $...
1
vote
2answers
43 views
Are $C^\infty$ Functions with all derivatives positive on [a,$\infty$),a$\gt$0 always made of exponential?
Are there any $C^\infty$ real functions except the exponential family and gamma function family which has all the derivatives of same sign on an interval [a,$\infty$) with a$\gt$0 ?
I speculate the ...
1
vote
0answers
17 views
Regarding the condition of Carlson's theorem
Assume $f$ is entire. I want to know if
$$|f(z) |<Ce^{\tau |z|} \: \& \: |f(iy)|<Ce^{c|y|}, y\in \mathbb{R}$$
for some constants $C,\tau$ and $c<\pi$, is equivalent to
$$ |f(z) |<Ce^...
1
vote
1answer
26 views
If a real multi-variable function is analytic along all analytic curves passing through $0$, is it real-analytic?
Given $f(x):\mathbb{R}^n\rightarrow\mathbb{R}$, if for any curve $\gamma:[-\epsilon,\epsilon]\rightarrow\mathbb{R}^n:t\mapsto(\gamma_1(t),\ldots,\gamma_n(t))$ such that $\gamma(0)=0$ and each $\...
2
votes
2answers
73 views
Why do we take $(1+z)^{\alpha}$ as $e^{\alpha \operatorname{Log}(1+z)}$?
Let $\alpha$ be a complex number. Show that if $(1+z)^{\alpha}$ is taken
as $e^{\alpha \operatorname{Log}(1+z)}$, then for $|z|< 1$
\begin{equation*}
(1+z)^{\alpha} = 1 + \frac{\alpha}{1}z +
\...
0
votes
2answers
30 views
$f$ is constant complex function if $Af(z) +B \overline{f(z)} = 0$ for some $A$ and $B \in \mathbb{C}$
Let $U \subset \mathbb{C}$ be a open convex set and $f : U \to C$ a analytic function, such that there exist $A, B \in \mathbb{C}$ and
$ Af(z) +B\overline{f(z)}=0$
Show that $f(z)$ is constant.
I ...
1
vote
2answers
34 views
Analytic functions on $\mathbb{Q}$
$\mathbb{Q}$ has the topology induced from $\mathbb{R}$, therefore it is in principle possible to talk about power series and define analytic functions on $\mathbb{Q}$ to be power series (with ...
1
vote
1answer
38 views
Prove $\int_{1}^{\infty}\widetilde{B}_{2n+1}(x)x^{-(s+2n+1)}dx$ is analytic
Given $B_{n}(x)$ satisfies
\begin{equation}
\frac{ze^{xz}}{e^z-1}=\sum_{k=0}^{\infty}\frac{B_k(x)}{k!}z^k
\end{equation}
which is call Bernoulli polynomial. And $\widetilde{B}_{n}$ is 1-...
1
vote
0answers
30 views
Coefficients of the composition of an analytic function
Suppose we have a function $f(x):\mathbb{R}\rightarrow \mathbb{R}$ which is analytic. We can write its Taylor expansion around $x=0$ as: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \dots$.
The function $f^...
0
votes
2answers
27 views
Lie groups and orthogonal group
Are orthogonal groups are lie groups? I think parameter space points corresponds to elements with determinant -1 break analytic property of lie groups , what is the general condition to check a ...
1
vote
1answer
31 views
Extend $f(z)=\frac{1}{z^n +z^{n-1}+…+z^2 + z^{-n}}+\frac{c}{z-1}$
find $c $ such that $ f(z)=\frac{1}{z^n +z^{n-1}+...+z^2 + z^{-n}}+\frac{c}{z-1}$ can be extended to be analytic at $z=1$ , when $n\in \mathbb{N}$ when $n$ is fixed.
The given function I write it ...
0
votes
1answer
20 views
$f(z)=\sum_{n=1}^{\infty} a_n(z-z_0)^n$ such that $\sum_{n=0}^{\infty} f^n(a)$
Let $f(z)=\sum_{n=1}^{\infty} a_n(z-a)^n$ such that $\sum_{n=0}^{\infty} f^n(a)$ converges then is it necessary that $f(z) $ is entire.
I have tried this by giving counter example but all in vain , ...
0
votes
2answers
43 views
Prove/disprove the existence of an analytic map using Identity Theorem [closed]
I want to prove/disprove the following statement using Identity Theorem.
Does there exists an analytic map $f: \Bbb C \to \Bbb C$ such that $f(z)=z$ for all $z$ such that $|z|=1$ and $f(z)=z^2$ ...
4
votes
0answers
159 views
Analyticity of this function $\sqrt{\coth^2(a\ z) + \coth^2(b\ z) - c}$
I want to determine the domain of analyticity of this function:
$$f(z) =\sqrt{\coth^2(a\ z) + \coth^2(b\ z) - c}$$
And $$c \in ]0,1]$$
Where $$a,b \in \mathbb{Z} - \mathbb{Z}^+$$ and $a , b$ finite ...
0
votes
1answer
35 views
Analytic function need not have a primitive
I am told that an analytic function need not possess a primitive in its domain of analyticity. However, if $f$ is analytic in the disk $\text{B}(a,r)$, then it has a primitive in that disk. Suppose $f$...
1
vote
0answers
26 views
Using the Cauchy Principal Value along a differentiable contour?
I am looking at the following problem from Marsden and Hoffman:
Let $ f ( z ) $ be analytic inside and on a simple closed contour $ \gamma $. For $ z_0 $ on $ \gamma $, and $ \gamma $ differentiable ...
0
votes
1answer
36 views
If a real analytic function is zero at a point in a topological space, how does one prove that it is zero everywhere in that space?
I understand that an iterative argument can be used here, where one considers the points in a neighbourhood of the point at which the function is zero. However, I do not know how to go about proving ...
1
vote
1answer
18 views
Verification of Alternate Proof for Identity Theorem in Conplex Analysis.
Can't we prove Identity Theorem like this.
IDENTITY THEOREM.
Consider a function which is analytic on an open connected domain $D$, if the set of zeros of $f$ has a limit point in $D$ then $f=0$ ...
0
votes
3answers
57 views
Is there an analytic function $f$, such that $ f(\frac1n)=f(-\frac1n)=\frac1{2n+1}$, for all $n \in \mathbb N$?
Let $ f:\{z|\; |z| \lt 1\} \rightarrow \mathbb C $ be a non constant analytic function. Which of the following conditions can possibly be satisfied by $f$ ?
$f(\frac{1}{n})=f(\frac{-1}{n})=\...
0
votes
1answer
23 views
Generalizing an implicit function theorem for formal power series
This exercise is from Greuel & Lossen & Shustin's Introduction to Singularities and Deformations, Exercise 1.2.5.
Let $f\in\mathbf{C}\langle \mathbf{x},y\rangle$, where $\mathbf{C}\langle \...
4
votes
1answer
38 views
If an analytic map $f$ has “many” values in a negligible set $B$, does $\text{Image}(f) \subseteq B$?
Let $k>n$ be positive integers, and let $f:\mathbb R^n \to \mathbb R^k$ be a real-analytic map (i.e. every component of $f$ is a real-analytic function).
Suppose that we have a measurable ...
1
vote
0answers
46 views
$p$-adic analytic function bounded implies coefficients bounded?
Let $K$ be a complete valued subfield of $\mathbb{C}_p$. Let $\mathcal O=\{z \in \mathbb C_p \colon \vert z \vert \leq 1\}$ be the ring of integers in $\mathbb C_p$ and $\mathfrak m=\{z \in \mathbb ...
1
vote
1answer
43 views
Is there an analytic function (on the open unit disk) satisfying $\forall k \in \mathbb{Z}^+\left[f(\frac1{2k})=f(\frac1{2k+1})=\frac1{2k}\right]$? [duplicate]
Although I could be wrong, I suspect the question in my title will be answered in the negative.
Let $f:D \to \mathbb{C}$, where $D$ is the open unit disk. Assume that $\forall k \in \mathbb{Z}^+\left[...
0
votes
0answers
33 views
real analytic inverse function theorem
I have a two part question about real analyticity and whether I have a an error in reasoning.
Suppose I have a multivariate holomorphic mapping $f \colon U \to \mathbb{C}$, where $U$ is an open set in ...
0
votes
1answer
45 views
Estimate of Taylor coefficients [closed]
I have a real function f(x) different from zero and not proportional to $\exp(x)$ analytic at origin, $$ f(x)=\sum_{n=0}^{+\infty}c_n x^n,\ \forall x:|x|<\delta $$
To derive a result, I should ...
1
vote
2answers
38 views
Analytic function vanishes in an open unit disk. Show it vanishes identically.
$f$ is an analytic function that vanishes in a unit disk which is a subset of domain $D$. Show that it vanishes all over $D$.
This was one of our quiz questions and I think we need to assume that $D$ ...
0
votes
1answer
67 views
Laurent series problem on $f(z)=\frac{ z }{ z^2-z-2 }$
I have problems with computing Laurent series of the function $f(z)=\frac{ z }{ z^2-z-2 }\quad$ in the ring centered in $0$ containing point $1+i$. I also have to find the radius of convergence of ...
0
votes
0answers
64 views
If v is harmonic conjugate of u, then the harmonic conjugate of $3u^2 − v^3$ is harmonic conjugate of
Attempt:
I tried to use cr-equations,but integration becomes clumsy. Also, I tried to form another holomorphic function which have real part equal to $3u^2 − v^3$ using any other holomorphic functions,...
1
vote
0answers
50 views
$\int_0^{+\infty} f(x) dx = - \lim_{a\rightarrow 0^+} \frac{1}{a} \sum Res_{\Bbb{C} \setminus \Bbb{R}^+} (f(z)z^a)$?
I find integrals on the real axes computed by complex number techniques and looking for a generalization working just on the positive semiaxes. I tried to put together this general result, but I am ...
1
vote
3answers
28 views
Confusion about relationship between continuous and real analytic functions
By Weierstraß aprroximation theorem, every continuous function can be $\epsilon$-approximated by a polynomial, i.e., if $f:[a,b]\to\mathbb{R}$ is a continuous function, then for all $\epsilon>0$ ...
0
votes
0answers
13 views
What's the product of Bergman and Dirichlet functions on the disk?
Under what conditions can we claim that the product of a Bergman space function and a Dirichlet space function belongs to the Hardy space $H^2$ (or any $H^p$ with $p>0$), all on the unit disk $\...
0
votes
0answers
21 views
Convergence of Power Series and Power Series Solutions of ODE
A function which has a convergent power series expansion about a point is called analytic at that point. A function may not be analytic at some points but analytic every where else. This means that ...
0
votes
3answers
74 views
Show that f is constant. [duplicate]
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function and let $a \in \mathbb{R}$. Show that if $\Re f(z) \geq a$ for all $z \in \mathbb{C}$, then $f$ is constant.
Now I know that I need ...
0
votes
1answer
24 views
Some questions regarding complex analysis
I recently came across the following problem
Pick out the true statements:
There is a bijective analytic function from $\Bbb C$ to the upper half plane $\Bbb H$
There is a non-constant ...
0
votes
1answer
42 views
Connectedness of the domain in Identity Theorem
Let $f$ be a complex-valued holomorphic function defined on an open set $\Omega\subseteq\mathbb{C}$, $f:\Omega\rightarrow\mathbb{C}$, which is not identically zero.
Let $S=\{a\in \Omega :f(a)=0\}$=...
1
vote
2answers
68 views
Complex Analysis, find all analytic functions
Find all analytic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that
$|f(z)-1| + |f(z)+1| =4 $ for all $z \in \mathbb{C} $ and $f(0) = \sqrt{3} i$
I understand that the given equation ...
3
votes
1answer
84 views
Does anyone knows how to average function in three dimensions
$$
\begin{split}
\frac{dr}{dt} &= -\epsilon \sin^2(\theta)z \\
\frac{d \theta}{dt} &= -1-\epsilon \cos\theta \sin\theta z\\
\frac{dz}{dt} &= \epsilon(r^2-T)
\end{split}
$$
...
2
votes
2answers
106 views
Integral of real part of $z$ around the unit circle
What is the result of integrating the real part of z (a complex number) anti clockwise around the unit circle?
At first glance, I couldn't identify any points within the circle where analyticity ...
2
votes
1answer
52 views
How to justify this statement?
Let $f$ a holomorphic function in $\mathbb{D}(0,1)$ such that $f(z)=\sum_{n\ge 0}a_n z^n$. Then apparentely we can deduce that for all $r\in [0,1[$, $n\ge1$ : $\int \limits_{0}^{2\pi} \overline{f(r\...
0
votes
0answers
20 views
Analytic contiunation
this is more of a broader question.
Say I have an analytic complex function $f$ that is defined on the open unit circle, but I know that the limit of $f$ when $|z|$ approaches 1 is 0. Can you define ...
0
votes
1answer
23 views
Is there a “monotonicity” property for analytic continuation?
If I have two complex functions defined by power series $A(z) = \sum a_n z^n $, $B(z) = \sum b_n z^n$ with $|a_n| \ge |b_n|$ for all $n$, and I know that $A$ converges in some set $U_1$ and defines a ...
0
votes
1answer
39 views
Different ( Is it equivalent?) hypothesis for stronger version of Goursat's theorem
Ahlfors, states that
Theorem: Let $f(z)$ be analytic on the set $R'$ obtained from a rectangle $R$ (interior and on the boundary of the rectangle) by omitting a finite number of interior points $a_i$...
1
vote
0answers
29 views
Existence of the limit of a bounded analytic function
Let $X$ be a Banach space and let $U\subset\mathbb{C}
$ be open. If we have an analytic function $f:U\setminus \left\{ 0\right\}
\rightarrow X$ such that $\underset{x\in U\setminus \left\{ 0\right\} }{...
0
votes
1answer
39 views
How is it possibile for a complex analytic function not having a derivative in the region in which it is analytic?
There is a question asking if the function $f$, which is defined by $$f(z) = \frac{\bar{z}^2}{z},\quad z\neq 0$$ and $f(0) = 0$, is analytic at $z = 0$ and if there is $f'(0)$.
The answer is ...
