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

1
vote
1answer
208 views

Analytic continuation of complex square root

This example comes from the book by Elias Wegert: Visual Complex Functions. Consider the function $f(z):=z^{1/2}$ for $z \in \mathbb{C}$ with $|z - 1| < 1.$ For these $z$, the function can be ...
5
votes
1answer
103 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
4
votes
1answer
95 views

Commutativity of integration and Taylor expansion of the integrand in an integral

I am baffled with a seemingly a straightforward problem. Suppose we are given the following integral: \begin{equation} f(a)\,=\,\int_{0}^{\infty} \frac{x^4}{x^4+a^4} e^{-x}, \end{equation} and we ...
1
vote
1answer
193 views

$f $ is analytic and maps the unit disk to itself. Prove that $|f'(0)|\leq1- |f(0)|^2 $

I am having difficulties with the following problem: $\bf Given$: $f $ is analytic and maps from unit disk to itself. $\bf Prove:$ $|f'(0)|\leq1- |f(0)|^2 $. For some reason (unclear to me) it ...
1
vote
1answer
36 views

Real analyticity of $\sqrt{x}\coth(\sqrt{x})$ : function that is decreasing with resp. to derivation

I'm trying to show that $\frac{\sqrt{x}}{\tanh(\sqrt{x})}$ is real analytic at $x=0$ (with the principal determination of $\sqrt x$). Apparently (i.e. thanks to graphs with Maple) $n \mapsto ...
1
vote
1answer
52 views

When does analytic continuation respect functional equation

This is a subtle point about analytic continuation. Let $\Gamma(s)$ be the analytic continuation of $\gamma(s) := \int_0^\infty e^{-t}t^{s-1}dt$ to $\Bbb C \setminus \Bbb Z_{<0}$, the latter ...
2
votes
2answers
235 views

Show that the Cauchy integral formula implies the Cauchy-Goursat Theorem

I'm struggling with this question, the integral formula states: $$f(z_0) = \frac{1}{2\pi i} \int_{C}\frac{f(z)}{z-z_0}\,dz$$ and the Cauchy-Goursat theorem states: If $f$ is holomorphic in a simply ...
2
votes
2answers
88 views

$f$ analytic and $|f|$ a function of $|z|$

Suppose $f$ is analytic inside the unit disc and that $|f(z)|$ depends only on $|z|$. Prove that we can write $f(z) = Cz^N$, for all $z$ in the disc. In the suggested proof, it is stated like it's ...
0
votes
2answers
73 views

Show that $\log|\sin(z)|$ is the real part of a holomorphic function

$D$ is a connected, simply connected domain with $\sin(z)$ never zero on D. Show that $\log|\sin(z)|$ is the real part of a holomorphic function. My question is: how to show $\sin(z)$ maps a simply ...
5
votes
1answer
194 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
0
votes
1answer
39 views

Analytic functions in two variables

Let $f$ be an analytic function in two complex variables. It is well known that we can expand $f$ in a convergent series of two variables. Can we separate the variables in such a manner that $f$ ...
2
votes
2answers
171 views

Behavior of holomorphic functions on the boundary of the unit disk

$\textbf{Problem.}$ Suppose $f$ is holomorphic on the unit disk $\mathbb{D}$. Show there are points $a_n\in \mathbb{D}$, $a\in \partial \mathbb{D}$, and $b\in \mathbb{C}$ such that $a_n\to a$ and ...
21
votes
1answer
573 views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its ...
2
votes
2answers
99 views

Are analytic functions injective?

Let $f$ be analytic on the whole all of $\mathbb C$. Assume that $\mathrm{Re}\, f \ge 0$. What can we say about $f$? I'm thinking $f$ has got to be constant, since otherwise it would map the entire ...
3
votes
1answer
67 views

Prove f is analytic and periodic

Suppose that there are entire functions $\{f_n\}$ so that for all complex numbers $x+iy$ $$\sum_{n=1}^{\infty} |f_n(x+iy)|^{\frac{1}{n}} \leq e^x$$ Show that $f(z)=\sum_{n=1}^{\infty} f_n(x+iy)$ is ...
3
votes
1answer
148 views

for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if ?? CSIR - June $2013$

Question is : for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if $p(z)$ is Constant $p(z)q(z)$ is Constant $q(z)$ is Constant ...
1
vote
1answer
104 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 ...
5
votes
1answer
156 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 ...
0
votes
1answer
39 views

Specify analytic function

How can I check that function $f(z)=z^3+z-1$ is analytic or not without Cauchy-Riemann equations? $(z\in\Bbb C)$
4
votes
1answer
104 views

Odd and even square roots of $z^2-1$

This is a very interesting exercise (provided that it is correct). Find two holomorphic functions $\,f_1: \Omega_1\to\mathbb C$ and $f_2:\Omega_2\to\mathbb C$, which are both square roots of $z^2-1$, ...
3
votes
1answer
130 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 ...
0
votes
1answer
44 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 ...
0
votes
1answer
46 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 ...
0
votes
1answer
47 views

Suppose $f \in C^{\infty}(\mathbb R)$ and $\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0.$ Show $f$ is analytic on $\mathbb{R}$.

Suppose that $f \in C^{\infty} (-\infty , \infty)$ and that $$\lim_{n \to \infty} \frac{1}{n!} \int_0^a x^n f^{(n+1)}(a-x)dx=0$$ for all $a\in \mathbb{R}$. Prove that $f$ is analytic on ...
0
votes
1answer
78 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 ...
5
votes
1answer
119 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 ...
1
vote
1answer
93 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 ...
1
vote
0answers
28 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 ...
0
votes
1answer
45 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 ...
2
votes
1answer
141 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. ...
1
vote
1answer
135 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 ...
6
votes
1answer
312 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 ...
19
votes
2answers
585 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 ...
0
votes
1answer
60 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 $C^\infty$ ...
0
votes
1answer
72 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.$A$ $B$ $X$ $Y$ ...
3
votes
2answers
176 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 ...
4
votes
2answers
378 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 ...
2
votes
0answers
102 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 ...
4
votes
1answer
153 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 ...
1
vote
1answer
76 views

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 $z\in D$. ...
2
votes
1answer
55 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 : ...
0
votes
4answers
97 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?
0
votes
2answers
203 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 ...
0
votes
1answer
39 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 $D_{f}$. ...
2
votes
1answer
43 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 ...
2
votes
1answer
184 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 ...
3
votes
3answers
309 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 $\sqrt{C}$?
3
votes
0answers
48 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 ...
1
vote
0answers
104 views

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 $z^3+2i=0$ ...
2
votes
2answers
194 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 ...