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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17 views

Is this function analytic: $F(x+iy)=\frac1{\pi}\int_{\mathbb R}\frac y{(x-t)^2+y^2}\,f(t)\,dt$.?

Let $f:\mathbb R\to\mathbb R$ such that $f(t)=0$ if $|t|\leq 1$ and $f(t)=|t|^\lambda$ if $|t|>1$. Here $\lambda<0$ is a constant. We consider the following function on $\mathbb C_+=\{x+iy:\;x,...
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89 views

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. [closed]

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. For what functions do we have equality? I'm not sure ...
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35 views

Analytic, non- constant complex function has finitely many zeros inside the disk $D(0, R)$ for all $R > 0$

I'm learning about complex analysis and need help with the following problem: Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and non-constant. Show that for every $R > 0$, the complex function $...
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42 views

Application of Schawarz lemma??

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. We have $|f(z)|\leq 1-|z|$ for all $z$ ...
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16 views

Mobius transformations

Suppose f is a continuous function on the extended complex plane which is analytic except possibly at one point and maps lines and circles to lines and circles. Does it follow that f is necessarily a ...
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26 views

Interior Uniqueness: Does there exist an analytic function on a neighborhood of $z=0$ that satisfies the following?

I am faced with the following problem: Does there exist a function that is analytic on a neighborhood of $z=0$ and satisfies the following condition for every positive $n$: (a) $f(1/n)=f(-1/n)...
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29 views

Local Uniform Convergence and Composition

I've been sitting down can't quite tell if this is true or not, but I suspect that it should be. Edit: Suppose that $\Omega$ is a open, connected subset of $\mathbb{C}$, and suppose that $(f_n) \...
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42 views

function analytic in the entire complex plane is constant

Let the function $f$ be analytic in the entire complex plane and suppose that $\frac{f(z)}{z}\rightarrow 0$ as $|z|\rightarrow \infty$. Prove that $f$ is a constant. As $\frac{f(z)}{z}\rightarrow 0$ ...
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209 views

Laurent expansion of $\frac{1}{\sin z}$

Question is a fully solved exercise in Gamelin's complex analysis. Exercise : Consider the Laurent series expansion for $\frac{1}{\sin z}$ that converges on the circle $\{|z|=4\}$. Find the ...
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48 views

Why is this point analytic?

Suppose you had the function $$ p(x) = \frac{\sin(x)}{x} $$ I know, from other material online, that this point is analytic at the point $x=0$. However, my understanding was that a point of a function ...
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31 views

How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
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13 views

How is it possible to find all singular (non-analytic) points of a differential equation?

Suppose we had a second order differential equation of the form $$ y'' + p(x)y' + q(x)y = 0 $$ I know that a point $x=x_{0}$ is said to be ordinary if $p(x_{0})$ and $q(x_{0})$ can be expressed as an ...
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39 views

Laurent expansion

Question is to write Laurent expansion of $f(z)=\dfrac{1}{(z-2)(z-1)}$ in the annulus $1<|z|<2$ based at $z=0$ I am aware of the method of partial fractions and writing expansions for both $\...
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1answer
64 views

Power of a function is analytic [duplicate]

Question is : Show that if $f(z)$ is continuous function on a domain $D$ such that $f(z)^N$ is analytic on $D$ for some integer $N$ then $f(z)$ is analytic on $D$.. For some time i was wondering ...
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15 views

The existence of the roots of an holomorphic function on an open connected domain

Let $U$ be an open connected domain and $D$ be an open disk such that the closure of $D$ is a subset of $U$. Suppose $f\in H(U)$, i.e., $f$ is holomorphic in $U$, and that $f$ is not constant. Show ...
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20 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
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1answer
38 views

Evaluate $\int_{|z|=1}\frac{|dz|}{|z-a|^2}$ using Cauchy's Theorem.

Question: Evaluate $\int_{|z|=1}\frac{|dz|}{|z-a|^2}$ using Cauchy's Theorem. My attempt: So, Cauchy's theorem for derivatives tells us that if $f$ is holomorphic in an open set $\Omega$, and $D$ is ...
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1answer
29 views

Let the function $f$ be analytic in $C$, real valued on $R$, and $\Im f(z) > 0$ in the upper half-plane. Prove that $f'(x) > 0$ for $x \in R$.

Question: Let the function $f$ be analytic in the entire complex plane, real valued on the real axis, and of positive imaginary part in the upper half-plane. Prove that $f'(x) > 0$ for $x \in R$. ...
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1answer
16 views

Analytic function on the annulus $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ s.t. $C_1 \to C_1$, $C_4 \to C_2$?

Question: Does there exist an analytic function mapping $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ and taking $C_1 \to C_1$, $C_4 \to C_2$, where $C_r$ is the circle of radius $r$...
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46 views

A single analytic function that can approximate all others

The problem in it's entirety is this: Given some simply connected domain $U$ then show that there is a function $g\in\mathcal{O}(\mathbb{C})$ such that for any given $f$ there exists a sequence $\...
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29 views

Let $f(z) = u(x,y) + iv(x,y)$ be analytic in $\Omega$, suppose that $v(x,y) = e^{-y}(y\cos x -x \sin x)$, find $f(z)$.

Question: a) Let $f(z) = u(x,y) + iv(x,y)$ be analytic in $\Omega$, suppose that $v(x,y) = e^{-y}(y\cos x -x \sin x)$. Find $f(z)$. b) Let $f(z), g(z)$ be analytic in an open, connected domain $\...
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53 views

Approximating polynomials over $\mathbb{C}$ with an entire function

Given a series of polynomials $p_{n}$ and a series of non-intersecting balls $B_{n} \subset \mathbb{C}$ show that there exists some function $f \in \mathcal{O}(\mathbb{C})$ such that $lim_{n \...
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1answer
133 views

Analytic function on unit disk has finitely many zeros

I am studying complex analysis from Theodore Gamelin's text and Exercise 1 of chapter IX.2 says that if $f$ is analytic inside the open unit disk and continuous on its boundary that satisfies $|f(z)| =...
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48 views

Some proofs about the infinite products; the series and the analyticity

Definition A.1.4. The infinite product $\prod_{n=1}^{\infty}(1+a_{n}(x))$, where $x$ is a real or complex variable in a domain, is uniformly convergent if $p_{n}(x)=\prod_{m=k}^{n}(1+a_{n}(x))$ ...
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49 views

Analyticity of solutions to the heat equation

Let us look at solutions to the linear heat equation on $\mathbb{R}$: $$ u_t = u_{xx}.$$ Is it true that solutions to the equation with nice enough initial datum are analytic after a certain time $T &...
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35 views

Is the solution of a polynomial an analytic function on the polynomial parameters?

Be $\mu(z_1, \ldots, z_L)$ the only positive real solution to the equation \begin{equation} \sum_{l=1}^L z_l \mu^l = 1 \end{equation} With $z_1 = 1$, $z_l \geq 0 \forall l$. Clearly, varying the ...
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23 views

Prove: Show $\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$ for $f(z)=z^n$ and any $n \in N$

Prove that if $f:\mathbb{D} \to \mathbb{D}$ (where $\mathbb{D}$ is the unit disk) is given by $f(z)=z^2$, the for all $z \in \mathbb{D}$, we have $$\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$$ ...
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30 views

Holomorphical extension to the Annulus

Let $D=\{z:1<|z|<2\}$ and $f$ is holomorphic on $D$. Suppose that f has a primitive $f_1$ on D and $f_1$ also has a primitive $f_2$, etc for every $n$ $f_n$ has a primitive $f_{n+1}$ in $D$. How ...
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1answer
16 views

Suppose $\{f_n\}$ is a sequence of analytic in a region $D$

suppose $\{f_n\}$ is a sequence of analytic in a region $D$ such that $|f_n(z)|\leq M_n$, where $\sum_n M_n<\infty$, and $\lim_{z\to z_0}f_n(z)=L_n$. Show that if $p=p(z)\to\infty$ as $z\to z_0$, ...
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48 views

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges. Is $$F(z)=\int_0^\infty g(t)\sin(zt)dt$$ analytic? And if so, in what region? My attempt: $$F(z)=\int_0^...
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91 views

Simply connected domains and complex logarithms

While studying Complex Analysis from my professor's notes I came across the following theorem. A demain $D$ in the complex plane is simply connected if and only if any analytic function $f(z)$ on $...
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57 views

If $f^{\prime}(z)=g^{\prime}(z)$, then $f(z)=g(z)+c$

Suppose $f$, $g:G \to \mathbb{C}$ are defined on a domain $G \subseteq \mathbb{C}$ and are differentiable on $G$. Then, if $f^{\prime} = g^{\prime}$, then $f = g + C$ for some constant $C$. I ...
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1answer
37 views

$f(1/n)=1/n$ implies $f(z)=z$

If $f:\mathbb{C}\to\mathbb{C}$ is entire and satisfies $f(1/n)=1/n$ then $f(z)=z$. I am trying to prove this. What I have so far: by continuity $f(0)=0$. Since $f(1)=1$, it feels tempting to show ...
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2answers
81 views

If the real part of $f$ is bounded then $f$ is constant

It isn't too hard to show that if $f:\mathbb{C}\to\mathbb{C}$ holomorphic everywhere (entire) and $\Re (f)$ is bounded, then $f$ is constant: it suffices to consider $\exp f$, which is entire, and by $...
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20 views

Composition of analytic functions is analytic in Manifolds

My problem is in analytic manifolds.According to Cohn's book a function $f$ in a manifold $M$ is analytic at $p \in M$ if it can be expressed as a power series of $\sigma(p)=(x_{0})$. That means $f=...
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21 views

$D:=\{z \in \mathbb C: |z|<1\}$ and $f:D \to \mathbb C$ be an analytic function such that $|f(z)|\le 1-|z| , \forall z \in D$ , then $f=0$? [duplicate]

Let $D:=\{z \in \mathbb C: |z|<1\}$ and $f:D \to \mathbb C$ be an analytic function such that $|f(z)|\le 1-|z| , \forall z \in D$ , then is $f$ identically zero ?
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1answer
27 views

If $f,g$ real analytic and $\lim_{t \to t_0} f(t)/g(t)$ exists then $f/g$ is analytic

If $f,g$ are real analytic at $t_0$ and $\lim_{t \to t_0} f(t)/g(t)$ exists then is it true that $f/g$ with the limiting value filled in at $t= t_0$ is real analytic at $t_0$? I know the complex ...
14
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201 views

Is this function nowhere analytic?

One usually sees $f(x):=\exp\frac{-1}{x^2}$ as an example of a $C^\infty$ function that is not analytic, having one point of non-analyticity (the point $0$). The Fabius function is a canonical ...
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1answer
38 views

Finding analytic function

I am trying to solve the following problem, "Show that if $h(z)$ is a complex-valued harmonic function such that $zh(z)$ is also harmonic, then $h(z)$ is analytic." My approach was to calculate first ...
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1answer
28 views

Clearer definition of a singularity?

"A point $z$ is said to be a singularity of the function $F(z)$ if in the complex plane there exists no circle with center at $z$ within which $F(z)$ is analytic." Can someone describe this a little ...
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68 views

Taylor series expansion of $ f(x)=e^{-x^2}$

How to find Taylor series expansion of $f(x)=e^{-x^2}$ What I'm stuck at is proving that the error $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$$ of the expansion tends to zero.
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106 views

Which meromorphic functions are logarithmic derivatives of other meromorphic functions?

Let $f$ be a meromorphic function defined on the whole complex plane. Is there a characterization in terms of easier-to-test properties of $f$ whether or not $f=g'/g$ for some entire $g$? The ...
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0answers
55 views

Show that $f(z) = \ln r + i \varphi$ is differentiable in a neighborhood of $z_{0}$

I am faced with the following problem: Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood ...
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15 views

Analytic structures which induce same topology on $\mathbb{R}$

It an exercise in Cohn's book. Analytic structure on a Hausdorff space $M$ is a family of charts $\mathcal{F}$ satisfying At each point of $M$ there is a chart in $\mathcal{F}$ Any two charts of $\...
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1answer
50 views

Show that $g$ is analytic and discuss the properties of $g$

Let $f$ be analytic in $\overline{B}(0; R)$ with $f(0)=0$, $f'(0) \neq 0$ and $f(z) \neq 0$ for $0<|z| \leq R$. Put $\rho=\min\{|f(z)|:|z|=R\}>0$. Define $g: B(0; \rho) \rightarrow \mathbb{C}$ ...
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2answers
62 views

Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$

I'm trying to find an example of an analytic function $f$ satisfying the IVP $$ f'(z) = z\,f(z), \quad f(0) = 1, $$ and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to ...
4
votes
3answers
133 views

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a polynomial

I'm learning about complex analysis and need some help with this problem: If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a ...
0
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2answers
60 views

Does there exist an analytic function $f$ on $D(0,1)$ such that $f(z_n)=0$ for even $n$ and $f(z_n)=1$ for odd $n$?

Given that $(z_n)$ is a sequence of distinct points in $D(0,1)=\{z \in \Bbb C : |z| \lt 1\}$ with $\lim_{n \to \infty} z_n=0$, Can we find an analytic function $f$ such that $f(z_n)= \begin{cases} 0, ...
0
votes
2answers
39 views

A function that satisfies $|f(z)-\bar z|<0.9$ is not analytic in the unit circle.

I've came accros this excersize: Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$ suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where $\...
5
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0answers
69 views

Finding an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$

I encountered the following problem in the lecture note in my complex analysis class: Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$. ...