Questions tagged [analytic-functions]
For questions about analytic functions, which are real or complex functions locally given by a convergent power series.
941
questions
1
vote
0answers
30 views
How is this argument in this paper justifiable?
Eq. 10 in this paper https://arxiv.org/pdf/1806.04039.pdf
has an integral equation
\begin{equation}
\sigma(\lambda)=a_1(\lambda)+\frac{1}{N}a_1\left(\lambda+\frac{1}{g}\right)-\int_{\mathcal C} d\...
1
vote
2answers
60 views
Show that $f(z) $ is constant [duplicate]
Suppose that $f(z)$ is entire with $a$ and $b$ positive constants. If
$$f(z+a)=f(z+b i)=f(z)$$
for all $z$, show that $f(z)$ is constant .
Which theorem could I use ? Please give some hint thankyou
0
votes
1answer
19 views
Cauchy integral formula for a point outside the simple closed curve
Show that if $f$ is analytic inside and on a simple closed curve $C$ and $z_0$ not in $C$ then
$$ (n-1)!\int_C \frac{f^{(m)}(z)}{(z-z_0)^n}dz=(m+n-1)!\int_C \frac{f(z)}{(z-z_0)^{m+n}}dz$$
Since it is ...
0
votes
1answer
24 views
Analytic image of conjugate of a point
Let $f:C \rightarrow \{|z|\leq2021 \}$ be an analytic function. If $f(11+9i)=\frac{1}{2i}$, then find the value of $f(11-9i)$.
How should I proceed with the problem ?
-1
votes
0answers
23 views
Give the points at which the given functions will not be analytic [closed]
I don't know how to prove that a function is not analytic at a certain point help out, please
$$f(z) = \frac z {z - 3i}$$
Basically prove that it is not analytic at a "certain point"
0
votes
1answer
39 views
How to extend analytic function
Let $ \gamma:\left(a,b\right)\to\mathbb{C} $ be an analytic path (Meaning, near every $ t_{0}\in\left(a,b\right) $ the function $ \gamma $ has a Taylor expansion in $t-t_0 $ with positive radius of ...
0
votes
0answers
23 views
Prove that there doesn't exists a holomorphic function defined in the disk $D(0,2)$ which satisfies the following condition
Prove that there doesn't exist a holomorphic function defined in the disk $D(0,2)$ which satisfies
$$\frac{(-1)^n}{n}+e^{f(1/n)}=1 \space \forall n\in \mathbb{N}.$$
I am having trouble proving this. I ...
0
votes
1answer
26 views
A holomorphic function is identically zero
Problem statement:
Suppose $V ⊂ \mathbb{C}$ is open and connected such that $\mathbb{D}$ (the unit ball centered at $0$) is contained in $V.$ Let $f : V → \mathbb{C}$ is holomorphic, and $ \int_{\...
1
vote
0answers
23 views
$\forall x_0 ∈ (a,b) \exists c \in (a, x_0), d \in (x_0, b), c_0, d_0$, such that $\forall x \in (c,d) \exists \frac{|f^{(n)}(x)|}{n!} \leq c_0d_0^n$
$a < b$ are real numbers and the function $f : (a, b) \to \mathbb{R}$ is analytic.
Show that for each $x_0 ∈ (a, b)$ there exist $c \in (a, x_0)$ and $d \in (x_0, b)$ and positive constants
$c_0$, $...
0
votes
1answer
32 views
Schwarz Lemma application problem
Schwarz lemma says that "Let ${\displaystyle \mathbf {D} =\{z:|z|<1\}}$ be the open unit disk in the complex plane ${\displaystyle \mathbb {C} }$ centered at the origin, and let ${\...
0
votes
0answers
29 views
Proof $F^{\prime}(z) - \pi\bar{z}F(z)$ is orthogonal to $F(z) \in \mathcal{L}_{2}(\mathbb{C})$
I am struggling with a simple proof regarding orthogonal functions to elements of the Hilbert space $\mathcal{L}_{2}(\mathbb{C})$ equipped with the norm:
$$||{F}||^{2}_{\mathcal{L}_{2}(\mathbb{C})} = \...
0
votes
0answers
66 views
Type of Map $w(z)=\frac{1}{1+z^2}$
Define a map from the unit disc $|z|< 1$ $$w(z)=\frac{1}{1+z^2}$$
Question 1. Determine the properties that whether the map is analytic, conformal, isogonal, one one, onto or some other type.
How ...
0
votes
0answers
16 views
Coefficients of taylor series of an analytic function about different centers.
To be concrete, I would like to limit my question to the exponential function and real numbers. The power series centered at any point converges to the exponential because its radius of convergence is ...
2
votes
0answers
24 views
The intersection of zero loci of a family of real analytic functions
If $(f_i)_{i \in I}$ is a family of holomorphic functions on some open set $U \subset \mathbb C^n$ with zero loci $V_i$, then the intersection $\bigcap_{i \in I} V_i$ is complex analytic. Moreover, ...
0
votes
1answer
33 views
Prove/ give counter example: $\lim_{z\to\ w}f(z)=\infty\iff\lim_{z\to\ w}Re(f)=lim_{z\to\ w}Im(f)=\infty$
Let $f$ be a complex function defined in a deleted neighbourhood of w. Prove or disprove by counter-example:
$\lim_{z\to\ w}f(z)=\infty\iff\lim_{z\to\ w}Re(f)=lim_{z\to\ w}Im(f)=\infty$
I know this is ...
2
votes
0answers
36 views
Holomorphic injective function has non-zero derivative
I am reading E.Stein's Complex Analysis. I am stuck at some detail of his proof of holomorphic injective function has non-zero derivative.(I know this statement has some proof in the website, but I am ...
0
votes
1answer
41 views
Integral of an analytic complex function inside a closed disk
I am trying to solve an exercise from Busam, Freitag Complex Analysis 2nd ed. which also has a provided solution but I guess I did not understand it.
Here is the ...
2
votes
0answers
29 views
Why does rational dependence of second derivatives on lower-order derivatives imply the map is real-analytic?
Suppose $u(x,y),v(x,y)$ are smooth real-valued functions defined on an open connected domain $U \subseteq \mathbb{R}^2$, satisfying
$$
(\text{Hess}\, u)_{ij}=\frac{P_i(\cos v, \sin v, v_x,v_y,u_x,u_y)}...
2
votes
1answer
42 views
Well-definedness of complex $\sin^{-1}$
I am define the inverse of complex $\sin$. I know that $\sin$ the conformal map mapping the strip $\{(x,y):x\in(-\pi/2,\pi/2), y>0\}$ to the upper half plane as the composition of the conformal ...
4
votes
2answers
102 views
Why does rational dependence of $f'$ on $f$ imply that $f$ is real-analytic?
Let $f:(0,1) \to \mathbb{R}$ be a smooth function, satisfying the ODE
$$
f'=\frac{P(f)}{Q(f)},
$$
for some polynomials $P,Q$. We assume that $Q(f)(x)=Q(f(x))$ does not vanish on $(0,1)$.
How to prove ...
2
votes
1answer
44 views
Prove or disprove : The limit $\lim_{n\rightarrow\infty}z^3_n$ exists if and only if the limit $\lim_{n\rightarrow\infty}z_n$ exists
Let $z_n$ be a complex sequence.
The limit $\lim_{n\rightarrow\infty}z^3_n$ exists if and only if the limit $\lim_{n\rightarrow\infty}z_n$ exists
I think the statement is true. From the definition, an ...
2
votes
1answer
22 views
Question about analytic function at complex field
**Question: **
Is $f : C → C$ given by $f(z) = z^2 + z|z|^2$ differentiable at $z = 0$? If so, what is $f'(0)$?
Does $f^{n}(0)$ exist for $n ≥ 2$?
The first two questions are quite clear. It is ...
1
vote
2answers
58 views
Types of singularity that f may have at $0$
Suppose f is holomorphic on a punctured neighborhood $D^*=\{z: 0<|z|<1\}$ and $f(z) \notin [0, \infty) \subset \mathbb{R}$ $ \forall z \in D^*.$ What are the possible types of singularity that f ...
1
vote
0answers
30 views
$\lim f(t)^{g(t)} = 0^0 = 1$ if $f,g$ are analytic?
Wikipedia states that if $f,g$ are real analytic near $c$, and $f,g \to 0$ as $t \to c$, then along any path for which $f>0$ one has $f(t)^{g(t)} \to 1$. The proof Wikipedia cites is in German so I ...
1
vote
3answers
93 views
Are absolutely continuous functions analytic?
I am asking for a proof that any absolutely continuous function with absolutely continuous derivatives is analytic, once I am studying a function with the first property and 'd like to obtain the ...
0
votes
0answers
30 views
Analytic continuation and convergence of generating function
For a real sequence $(a_n)_{n\in\mathbb{N}}$ I am considering the generating function $s(z) = \sum_{n=0}^{\infty} a_n z^n$ and by some calculations I find that it has a representation $f:\mathbb{C}\to ...
0
votes
0answers
35 views
Analytical description of curve from numerical data
I need to find the analytcal solution of the red curve in the image.
I have already found the expression of the blue curve, that is:
$(1^2 + 2^2) \cos^2 \tfrac{\varphi}{2} + (0.5^2 + 1^2) \sin^2 \...
2
votes
1answer
50 views
Complex functions and real function [closed]
This is a brief part of an exercise related to differential geometry on the minimal surface Scherk. However I am stuck in an analytical part.
Let $$F(x,y)= \arg\left(\frac{z+i}{z-i}\right)$$ where $z =...
-1
votes
0answers
40 views
If $f(z)$ is analytic then $f(\bar z)$ is analytic iff $f$ is constant. [duplicate]
I need to prove this using the C-R equations. I am a beginner and I really don't know much about the methods of proof usually employed in such questions. Please help. Thanks in advance.
0
votes
1answer
46 views
Understanding Identity theorem proof in complex analysis
Identity theorem
Let $f$ be holomorphic functions on some domain $D\subset\mathbb{C}$, and let $S$ be the set of all zeros of $f$ which has a limit point in $D$. Then $f$ is identically zero in $D$.
...
1
vote
1answer
27 views
Analytic function with arbitrary, countable zero set
Let $S\subset\mathbb{C}$ be a countable set with no accumulation points. Does there exist an analytic function $f(z)$ such that $f(z_i)=0$ for all $z_i\in S$ and $f(z)\neq 0$ for $z\neq S$?
Work so ...
0
votes
0answers
13 views
Zeros of analytic function is at most countable [duplicate]
Is this statement true?
Each analytic function on an open set can have at most countable number of zeros.
Can you give a me proof please?
Thank your sir and Madam.
1
vote
1answer
40 views
Hilbert transform of x (or a sawtooth)
This has a simple curiosity value, but I was converting the czt() (chirp z-transform) function from Octave into (wx)Maxima and, while testing it, I used a simple ...
0
votes
2answers
75 views
Is the zero set of an analytic function closed nowhere dense?
Given a non-constant analytic function $f(x)$ on a domain $D \subseteq \mathbb{R}^n$. I want to prove that
$\mathcal{Z} = \{x \in \mathcal{D} |f(x) = 0 \}$
is closed nowhere dense.
I originally wanted ...
0
votes
1answer
28 views
Specific branch of complex logarithm
Find a branch of $\log(3z-2)$ which is analytic at $z=0$ and takes the value $\ln 2+\pi i$ there. Specify the domain on which this branch is analytic.
These are my thoughts. I know that $\log z$ is ...
0
votes
1answer
38 views
Ignoring higher order terms in computations involving real analytic functions
We have the following four analytic functions of a real variable:
\begin{align}
\lambda (\epsilon) &= 0 + \lambda_1 \epsilon + \lambda_2 \epsilon^2 + \ldots \\
\tilde{\lambda} (\epsilon) &= 0 ...
2
votes
1answer
61 views
Prove or disprove a proposition on Taylor series
Let $f(x) = \sum_{k\geq 0} a_kx^k$ be a real analytic function, converging on $-R < x < R$, whose $n^{th}$ Taylor polynomial is $f_n(x) = \sum_{k=0}^n a_kx^k$. If there exist $N\in\mathbb{N}$ ...
1
vote
1answer
76 views
The Zeta Function and the Prime Number Theorem
I an trying to learn more about the $\zeta$ function, and I stumbled upon this very interesting paper https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Riffer-Reinert.pdf and having a few ...
0
votes
0answers
43 views
Reason(s) why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$
I am studying complex analysis and I am confused about why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$. For a function to be analytic, it must be differentiable and single-valued. Obviously, $f'(z)=-1/...
1
vote
1answer
49 views
How is the solution of an ODE identically zero?
I am reading Sturmian theory on Edward Ince's classic Ordinary Differential Equation and I again come across the statement that's been baffling me:
Consider $\frac{d}{dx}\left(K\frac{dy}{dx}\right)-Gy=...
2
votes
1answer
30 views
Convergence of power series at the boundary
Consider the following complex power series
$$
\sum_{n\geq 1}{\frac{ni^n}{2^n}{z^{n-1}}}
$$
By the root test, I have concluded that the disc of convergence is $D:=D(0,2)$. Then, I would like to study ...
2
votes
2answers
47 views
Fixed Points in Mobius Transformation
Let $$w(z)=\frac{e^z-1}{e^z+1}; \ z=x+iy$$ and $$w=u+iv$$ Then show that the image of the $y$-axis in the domain is $v$-axis in the co-domain. Does this contradict the result that a Mobius ...
0
votes
0answers
23 views
A question regarding differentiability and the boundary of analytic functions
Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be ...
1
vote
1answer
50 views
Prime ideals in ring of real analytic maps correspond to closed submanifolds?
In affine algebraic geometry, prime ideals in a ring $R$ correspond to irreducible subschemes of $Spec(R)$. For a real, compact, smooth manifold $M$, each prime ideal in $\mathcal{C}^\infty(M)$ is ...
1
vote
0answers
25 views
Kuramoto model has analytic solution.
I read a paper about a Kuramoto model, In this paper Kuramoto model has analytic solution of any initial value $\Theta(0)=(\theta_1(0),\theta_2(0),...,\theta_n(0))$
and the Kuramoto model is
$$\dot{\...
0
votes
0answers
22 views
Why is the integral of the derivative of f over f equal to zero over closed curves? [duplicate]
I'm trying to show that if $f(z)$ is analytic an $|f(z) - 1| < 1$, then $\int_\gamma \frac{f'(z)}{f(z)}dz = 0$ over all closed curves $\gamma$.
Presumably, I need to show that this is an exact ...
0
votes
1answer
32 views
Why is the derivative of f equal to the sum of its partials along its components?
I was trying to understand the development of the solution in this answer, where $\overline{f(z)}f'(z)dz = (u - iv)(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y})(dx + idy)$. The ...
-1
votes
2answers
49 views
Show that $f(z) = z^n$ is an entire function [closed]
Show that $f(z) = z^n$ is an entire function where $n = 0,1,2...$
I can show that $z$, $z^2$ and $z^3$ are entire functions but what about $z^n$ how do you crack that one.
0
votes
1answer
36 views
Integrate $\oint_{C} \sec z \mathrm{d}z$
The literature I read says that
$$\oint_C \sec z \mathrm{d}z = 0$$
This is what I know;
From the Cauchy integral theorem $\oint_C f(z)\mathrm{d}z = 0$ if $f(z)$ is analytic in a simple connected ...
0
votes
0answers
68 views
Is the complex function $f(z) = \sec z$ analytic?
Please help.
Is $f(z) = \sec(z)$ analytic?
I do know that I have to test if the cauchy riemann equations hold. That is
$U_x = V_y$ and $U_y = -V_x$
But I have trouble expressing $f(z)$ in the form
$f(...
