Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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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\...
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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
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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 ...
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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 ?
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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"
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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 ...
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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 ...
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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_{\...
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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 $c_0$, $...
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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 ${\...
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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})} = \...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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)}...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 =...
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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.
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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 $D$. ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 $N\in\mathbb{N}$ ...
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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 ...
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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/...
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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=...
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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 ...
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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 ...
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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 ...
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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 ...
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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{\...
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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 ...
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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 ...
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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.
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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 ...
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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(...

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