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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79 views

The natural boundary of the modular $\lambda$ function is the real axis

I want to solve the following problem from Ahlfors' text: Show that the function $\lambda(\tau)$ introduced in Chap. 7, Sec. 3.4, has the real axis as a natural boundary. Here $\lambda(\tau)$ is ...
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105 views

(solution verification) the series $\sum z^{n!}$ has the unit circle as a natural boundary

I've tried to solve the following problem from Ahlfors' complex analysis text: If a function element $(f,\Omega)$ has no direct analytic continuations other than the ones obtained by restricting ...
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37 views

Specify analytic function

How can I check that function $f(z)=z^3+z-1$ is analytic or not without Cauchy-Riemann equations? $(z\in\Bbb C)$
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21 views

Is $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ a general analytic function?

I have an expression $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ where $c(k)\in\boldsymbol{C}$ and $k\in\boldsymbol{R}$. $f(z)$ is an analytic function, since it contains only non-negative ...
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95 views

Complex Analysis: Log Function

I want to approach this problem with maximum understanding of everything that is going on. I have the function $F(z)=\log(z^2+4)$, and I want to give a region in which it is analytic. I guess I ...
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41 views

Convergence of an analytic function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function. Let $R$ be the radius of convergence of the Taylor series centered at $a.$ For each $n \in \mathbb{N},$ let $M_n= \sup\{f^{n}(t) : t \in ...
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40 views

Proving that a metric on space of analytic functions is equivalent to compact convergence

Let $U\subseteq \mathbb C$ be open and $\mathscr A(U)$ consist of all analytic functions on $U$. I can easily prove that there exists a sequence $K_n$ of compact sets in $U$ so that ...
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63 views

convolution of measurable function with analytic function

Let $f$ be a bounded measurable function with support on the unit disk $\mathbb D \subset \mathbb R^2$ and let $g$ be an analytic function on $\mathbb R^2$. Is it true that the convolution $h = f ...
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97 views

Proof of the three-point characterization of holomorphy

This post on Math Overflow is looking for the source of the following theorem: Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ denote the open unit disk. A function $f : D \to D$ is holomorphic iff ...
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71 views

Specifying a holomorphic function by a sequence of values

Given a sequence $(z_n, w_n)$ of pairs of complex numbers such that $|z_n| \to \infty$ as $n \to \infty$, there exists a holomorphic function $f$ such that $f(z_n) = w_n$ for all $n$. Proof: By the ...
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26 views

When is meromorphic continuation possible?

Suppose I have an expression of the form $$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region ...
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42 views

Principal Logarithmic Question

Here is a question that is driving me insane: Show that $p.v \sqrt{z-1}\times p.v\sqrt{z+1}=-p.v.\sqrt{z^2 -1}$ for $Re(z)<-1.$(p.v. stands for the principal singular valued logarithmic ...
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1answer
119 views

To what extent is a function that is analytic on the unit disk determined by its boundary values?

Suppose we have a function that is analytic on the open unit disk. Suppose we have a continuous function on the boundary of the disk that maps each point on the boundary of the disk to its conjugate. ...
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1answer
128 views

Composition of continious and analytic map

Let $U,V,W\subset\mathbb{C}$ open and connected, $f:U\to V$ continous and $g:V\to W$ analytic and non-constant. If $g\circ f$ is analytic, does then $f$ have to be analytic as well? I guess the ...
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234 views

Extended Proof of the Theorem that a bounded analytic function is constant.

I am having trouble feeling convinced by my proof and more importantly - feeling confident in my working out. The question reads (a) Let $f$ be an entire function such that there exist real ...
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523 views

Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the ...
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58 views

Two $C^\infty$ functions which agree on a set containing an accumulation point, but do not agree on *any* neighborhood?

As I understand it, two analytic functions defined on $\mathbb{R}^k$ which agree on a set with an accumulation point must agree on a neighborhood; however, the same is not true of $C^\infty$ ...
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70 views

Can we get an analytical solution to this equation involving the Lambert W function?

Can we get an analytical solution to the variable $t$: $$H\left(1+W\left(A\exp\left(Bt\right)\right)\right)=1+W\left(X\exp\left(Yt+Z\right)\right)$$ $W(x)$ is the Lambert W function.$A$ $B$ $X$ $Y$ ...
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154 views

Using Taylor expansion for a smooth real function

I've come across the following problem in Cracking Mathematics Subject Test, 4th Edition by Steve Leduc, from Princeton Review. Let $f(x)$ be a function that has derivatives of all orders at every ...
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298 views

Where is $\operatorname{Log}(z^2-1)$ Analytic?

$\newcommand{\Log}{\operatorname{Log}}$ The question stands as Where is the function $\Log(z^2-1)$ analytic , where $\Log$ stands for the principal complex logarithm. My understanding is that ...
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92 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
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112 views

A question related to uniqueness principle theorem.

We know that the equation $ \sin^2z+ \cos^2z=1$ which holds $ \forall z \in\Bbb R$, also holds $ \forall z \in\Bbb C$. This is obvious under the shadow of following theorem: Uniqueness principle ...
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Identical complex functions.

Uniqueness principle theorem : If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$. ...
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54 views

Want to check analyticity of a series on a open disk.

How do we check the analyticity of a any power series? For example: How will we show that $$f(z):= \sum_{n=1}^\infty z^{n!}= z^1+z^2+z^6+z^{24}......+z^{n!}......$$ is anaytic on disk {$z : ...
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95 views

Determine all values of $z$ (if any) where $f(z) = e^{|z|}$ is analytic?

So, I proceeded with the Cauchy-Riemann equations after setting $z = x+ iy$ and so $f(z) = e^{\sqrt{x^2 + y^2}}$, then I got the Cauchy-Riemann equations, but how can I proceed after this?
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142 views

When a holomorphic function is identically 0

I'm trying to prove this theorem (Theorem 4.8, Chapter 2, page 52, Complex analysis by Stein and Shakarchi): Suppose f is a holomorphic function in a region $\Omega$ that vanishes on a sequence of ...
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1answer
37 views

Can we deduce that $f(b)=f(a)+(b-a)(Re(f′(z₁))+iIm(f′(z₂)))$?

Complex Mean Value Theorem. Let $f$ be a holomorphic function defined on an open convex subset $D_{f}$ of $ℂ$ (or we can assume that $f$ is entire). Let $a$ and $b$ be two distinct points in $D_{f}$. ...
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42 views

About argument and modulus of a complex function

Let $ϕ(s)$ be an analytic function that has zeros outside a simply connected domain $D$. The function $ϕ(s)$ can be written as $ϕ(s)=ϕ₁(s)+iϕ₂(s)$ and therefore it is given uniquely by the polar form ...
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172 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
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256 views

$|f|$ constant implies $f$ constant?

If $f$ is an analytic function on a domain $D$ and $|f|=C$ is constant on $D$ why does this imply that $f$ is constant on $D$? Why is the codomain of $f$ not the circle of radius $\sqrt{C}$?
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Upper bound for integral where the integrand is not bounded on [r,1].

I'm reading an article of S. Rohde (On an Estimate of Makarov in Conformal Mapping, 1988) and there is a lemma which says that for univalent function it is possible to find a subset of $\partial D$ ...
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44 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
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Determine where the function $f(z)=\operatorname{Log}(z^3+2i)$ is analytic.

I need to know if my intuition is correct here. Idea: I would use De'Moivre's formula to find all third roots of -2i and exclude one of these rays because these are the values that give $z^3+2i=0$ ...
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167 views

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is ...
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37 views

Does this property persists for the derivatives $f^{(k)}, k=1,2,..$

Let $f$ be a real non polynomial analytic function. Suppose that the function $f$ assumes arbitrarily large and arbitrarily small values, i.e., for all $K>0$, there are $a,b$ with $f(a)<−K$ and ...
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26 views

Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
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114 views

Complex Analysis Question About Analytic Functions

I have some questions about knowing where and where not functions are analytic. Here's a function, f(z)= $\frac{Log(z+4)}{z^2+i}$ -I know that this function is not defined for ...
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1answer
76 views

Analyticity of a real function on $[0,\infty)$

I'm struggling to understand the difference of the analyticity of a real and a complex functions. Consider the following real valued function which is a minimal example of a somewhat more involved ...
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1answer
32 views

Analyticity of Products

Assume we have two functions $f,g:\Omega\rightarrow\mathbb{C}$ that are analytic and a third function $h:\Omega\rightarrow\mathbb{C}$ with $f=g\cdot h$. Can one now show that $h$ is analytic as well? ...
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109 views

Is discontinuity along a line equivalent to branch cut?

Suppose I claim the analytic function $f(z)$ has a branch cut along the positive real line, how would one go on to prove this? Is it sufficient to prove that $f(z)$ is discontinuous across this line? ...
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Schwarz Reflection Principle for function complex on the real line

The Schwarz reflection principle is usually proved for function real on the real (or a subset of) line. I wonder if the same principle/theorem works for general analytic functions on the real line? ...
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Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...
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108 views

Applications of identity theorem to physics

Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the entire complex plane. So, just by knowing how the function behaves at a teenie-weenie open disc ...
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426 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.
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1answer
73 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 ...
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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?
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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 ...
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88 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 ...
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51 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} ...
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85 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 ...