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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3
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1answer
39 views

Mapping the region $\Gamma_{z}$ using the conformal map $ \omega=\frac{-2z}{z^{2}+1}$

Suppose we have an analytic function $$ \omega=\frac{-2z}{z^{2}+1}$$ and the region $\Gamma_{z}$ given by $$\Gamma_{z}:=\left \{ z \in \mathbb{C}| \Im \left ( z \right )\geq 0 \wedge \left | z \right ...
2
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1answer
34 views

Complex inequalities and constants

Let $f(z)$ be an analytic function. Show that if $|f(z)| > 1 + |e^z|$ then $f(z)$ is constant. I have no idea what to do, I've subbed $z= x+iy$ and got $|f(z)|>1+e^x$ but lost here.
0
votes
2answers
30 views

Finding analyticity

Given that f is analytic, under what conditions is $g(z)=\overline{f(z)}$ analytic? Does this explanation make sense? : $g'(z)=lim_{h\rightarrow 0} \dfrac{g(z+h)-g(z)}{h}=lim_{h\rightarrow ...
0
votes
1answer
33 views

If $F$ is analytic and injective on the unit disc and $B(F(0), |F'(0)|)\subseteq F(B(0,1))$, then $F(z)= F(0) + F'(0)z$.

Let $F$ be analytic and injective on $B(0,1)$. Show that if $B(F(0), |F'(0)|)\subset F(B(0,1))$, then $F(z)= F(0) + F'(0)z$. I have tried the following: Since $F$ is injective we know that ...
1
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0answers
28 views

Multisection of a Power Series Proof

Suppose that $$H_{N,k}(x)=\frac{x^ke^{\frac{-x}{N}}}{N^{k-1}k!\sum_{n=0}^{N-1}{w_N^{-nk}e^{\frac{w_N^nx}{N}}}}=\sum_{n=0}^\infty{A_n\frac{x^n}{n!}}$$ where $k\lt N, w_N=e^{\frac{2i\pi}{N}}$, and ...
0
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3answers
48 views

Show that there exists $g: \mathbb{\Omega} \rightarrow \mathbb{C}$ analytic such that $g(z)^n = f(z)$ for all $z \in \mathbb{\Omega}$

I'm learning about complex analysis and need some help with this problem: Let $\mathbb{\Omega}$ open, simply connected, $f: \mathbb{\Omega} \rightarrow \mathbb{C}$ analytic without zeros in ...
0
votes
1answer
42 views

Prove a function to be analytic by dominated convergence theorem

Given $f \in L^1$, prove that $$F(z) = \frac{1}{2\pi i} \int^\infty_{-\infty} \frac{f(t)}{t-z}\,dt$$ is an analytic function and $$F'(z)=\frac{1}{2\pi i} \int^\infty_{-\infty} ...
2
votes
1answer
28 views

Cesaro limit of analytic functions

Let $f_n$ be a uniformly bounded sequence of analytic functions on $\Omega\subset\mathbb C$. If $f_n(z)\to f(z)$ forall $z\in\Omega$, then by the Montel's theorem I know that the convergence is ...
0
votes
1answer
62 views

Is my idea of decomposing a meromorphic function into a sum of Laurent series correct?

We know that complex-analytic functions $f(z)$ agree with their power series representations on their domain of analyticity. If a meromorphic function has simple poles at $z_1, ..., z_m$ and ...
2
votes
1answer
28 views

If $f$ is analytic and $f(0)=a_0$ then prove that $|\lambda| \ge \frac{|a_0|}{M}$.

Let , $f$ be non-constant holomorphic function in a nbd. of $\bar{\mathbb D}$ with $f(0)=a_0$. Let , $\displaystyle M=\max_{z\in \mathbb D}|f(z)$. Let, $\lambda \in \mathbb D$ and $f(\lambda)=0$. ...
1
vote
2answers
56 views

How can I show that this meromorphic function is a rational function of two polynomials?

Here's my updated attempt: Write$$f(z) = \sum_{n=-1}^{\infty} a_n(z-z_1)^n + ...+\sum_{n=-1}^{\infty} m_n(z-z_m)^n+\sum_{n=+1}^{-\infty} \psi_n(z)^n$$ with the last series being an expansion about ...
1
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0answers
40 views

Is this result for entire functions with zeroes only at the origin… more basic than the Hadamard canonical product representation?

I just worked on a problem and was able to solve it pretty easily, using Hadamard's product representation. But I wonder whether the solution that I compared my work to doesn't actually use the ...
0
votes
0answers
10 views

Bounded analytic functon with small derivatives

A question from the theory of bounded analytic functions. Let $f$ be analytic in the circle $D: |z|<1$ and bounded in $D$ by absolute value by a constant $M>0$. We assume that $N$ derivatives ...
1
vote
1answer
35 views

Prove that if $\overline {D_1}\subset f(\Omega ) $ then $\overline {D_r} \subset f(\Omega )$ for some $r>1$

Let $f:\Omega \to \mathbb C$ be an analytic function such that $\Omega $ is an open set.Define $D_r=\{z:|z|<r\}$ for some $r>0$. Prove that if $\overline {D_1}\subset f(\Omega ) $ then ...
1
vote
1answer
33 views

Bounded holomorphic function, limsup, boundary

Let $f \in \mathcal{O}(D)$, $\ D$ is a bounded region in $\mathbb{C}$, be such that $$\limsup _{D \ni z \to z_0} |f(z)| < \infty$$ for any $z_0 \in \partial D$. Prove that $f$ is bounded. So we ...
2
votes
0answers
76 views

How can I conclude that f(z) is constant? [duplicate]

I'm struggling a bit to arrive at the conclusion that $f(z)$ is a constant. Suppose $f(z)$ is holomorphic on and inside the unit disk, and that it has no zeroes on the interior. Also, assume that ...
1
vote
1answer
24 views

Is this a continuity / connectedness argument or is it an orientation-preservation argument?

Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle. We know the fact that the upper half plane (UHP) ...
0
votes
1answer
29 views

When computing contour integration with sines and cosines in the integrand, must we always first look at Euler's formula?

For example, in computing $$\int_{Cr}\frac {\cos(z)}{(z^2+a^2)^2}dz$$ over a semi-circular contour, must I first look at $$\int_{Cr}\frac {e^{iz}}{(z^2+a^2)^2}dz$$ compute this integral first, ...
1
vote
1answer
32 views

How can I compute the residue at this order-2 pole?

The integral is $$\int_{-\infty}^{\infty} \frac {cos(z)}{(x^2+a^2)^2}dz $$ If I use an upper semi-circular contour, then there is an order-2 pole at $z=ia$. I am trying to expand the integrand in a ...
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0answers
16 views

Can an analytic function defined on a maximal torus be extended analytically to all the Lie group?

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?
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0answers
21 views

Regarding the Schwarz Reflection Principle, is getting the analyticity of f(z) on the real axis a consequence of the theorem itself,

or a consequence of Morera's Theorem? Basically, I want to be able to cite it correctly, e.g., can I say we have not only continuity of f(z) along $R$ (by assumption) but also it turns out that ...
1
vote
1answer
43 views

Distance preserving transformations of the complex plane

Show that the most general transformation fixing the origin and preserving distances is either a rotation, or a rotation followed by a reflection in the real axis, for a transformation $f: \mathbb{C} ...
1
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0answers
39 views

Show that an analytic function defined on unit ball with these properties does not exist

Show that there does not exist an analytic function $F : B(0,1) \to \mathbb{C}-\mathbb{Q} $ with $F(0)=i$ and $F(1/2)=-i$. I have already used the open mapping theorem to show that if we assume ...
1
vote
1answer
44 views

Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.

Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$. I had no clue, I was trying to use the facts that ...
6
votes
2answers
153 views

Determine the complex function $f\left ( z \right )$

The details provided are that the function is analytic and that its real part along the line $y=c x$ is constant. What conclusions can I draw from here? I think that this imposes $\frac{\partial ...
0
votes
1answer
60 views

Why real valued harmonic functions are holomorphic.

Let $f$ be a real valued harmonic function on $C,$ then Claim $g= \frac {\partial f}{\partial x} - i \frac {\partial f}{\partial y} $ as holomorphic and $h= \frac {\partial f}{\partial x} + i \frac ...
0
votes
1answer
49 views

Why is a harmonic conjugate unique up to adding a constant?

If $v$ and $v_0$ are harmonic conjugates of $u$, then $u + iv$ and $u + iv_0$ are analytic functions. Then $i(v - v_0)$ is analytic, but how does this imply $v - v_0$ is a constant function?
4
votes
2answers
86 views

Example of real analytic function

We were taught real analytic functions in class today. I am playing around trying to construct examples. I see exponential, sine, cosine and logarithmic functions (for $x > 0$). One function I am ...
0
votes
1answer
34 views

Proove $|f(z)|\leq M|z|$ inside unit disk if $f(0)=0$ and $|f(z)|\leq M$

The problem goes as follows: Assume $f(z)$ is analytic inside the unit disk $D=\{z:|z|<1\}$. Also, $f(0)=0$ and $|f(z)|\leq M$ in $D$. Proove that $|f(z)|\leq M|z|$ in $D$. When does ...
0
votes
1answer
49 views

Proving that a complex function is analytic, and finding its power series

Let $I \subseteq \mathbb{R}$ be an interval and $g: I \to \mathbb{C}$ continuous. Define $f: \mathbb{C} \backslash \overline{Im(f)} \to \mathbb{C}$ by $f(z) := \int_I \frac{1}{g(x) - z} dx$ (with ...
0
votes
1answer
42 views

Zeros of real analytic functions on $\mathbb{R}^n$

Consider a non-constant multivariate real analytic function $f$ on $\mathbb{R}^n$. My question is, can the zeros of $f$ be dense in $\mathbb{R}^n$? In one dimension, I know that they cannot be, as the ...
0
votes
1answer
34 views

Prove convergence of $\int^{\infty}_{0}\ t^{z-1}cos(t)dt$ and $\int^{\infty}_{0}\ t^{z-1}sin(t)dt$

For a complex analysis problem set I am trying to show that the integrals $$\int^{\infty}_{0}\ t^{z-1}cos(t)dt \quad and \quad \int^{\infty}_{0}\ t^{z-1}sin(t)dt $$ is convergent for ...
5
votes
0answers
87 views

$f(x+1)=f(x)+f(\alpha\cdot x)$

I try to find an analytic increasing function $f_\alpha$ ($0\le\alpha\le1$) from $\mathbb R$ (or $\mathbb R^+$) to $\mathbb R$ such that for all $x$ $$f_\alpha(x+1)=f_\alpha(x)+f_\alpha(\alpha\cdot ...
2
votes
3answers
51 views

Approximation of non-analytic function

I have a function which is of the form \begin{equation} f(x) = \frac{1 - x^{1/2} + x - x^{3/2} + \ldots}{1+x^{1/2} - x + x^{3/2} - \ldots}. \end{equation} Intuitively, I would assume that for small ...
0
votes
1answer
39 views

Proove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function.

The problem is as follows: Proove that $U(x,y) = x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function. where $f(z)$ is analytic such that ...
3
votes
1answer
23 views

Is sudden appearance of poles possible?

I was thinking about the following scenario: For $|\epsilon|<1$ let $f_\epsilon(z)$ be a meromorphic function such that $f_0(z)$ has a pole at the origin for $\epsilon \neq 0$, $f_\epsilon(z)$ ...
0
votes
2answers
63 views

Does there exist a holomorphic function with the following property?

Does there exist a holomorphic function $f$ defined over $D = \{ z : |z| < 1 \}$ such that $|f| \rightarrow \infty$ when $|z| \rightarrow 1$? My approach: If such an $f$ exists, then for a given ...
1
vote
1answer
25 views

analytic deformation of a compact set in the complex plane

Let $K$ be an uncountable compact set in $\mathbb{C}$ such that zero is a limit point of $\partial K$, and such that $|k|\leq 1$ for all $k\in K$. I would like to find an analytic function ...
0
votes
0answers
50 views

Mittag-Leffler theorem for real analytic functions

I just started reading about the Mittag-Leffler theorem which says that given an open set $G \subset \mathbb{C}$, and a sequence of distinct points $a_k$ in $G$ (without a limit point), if we denote ...
3
votes
1answer
38 views

Does Cauchy's estimate imply analyticity?

Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic. How does one prove this? Surely, if Cauchy's estimates ...
0
votes
0answers
23 views

Reasoning in extra poles from analytic continuation

Entertain the following situation: Let $w(z)$ be a complex function which is a single valued function of $z$ inside the unit circle $|z|<1$. The derivative $\frac{dw}{dz}$ has simple poles at the ...
2
votes
1answer
51 views

Is continuity of first partials required for analyticity?

Let's cast the complex function $f(z) = u(z) + iv(z), z = x+iy$, as the multivariable function $F(x,y) = U(x,y) + iV(x,y) ; x,y \in R$. Thus, $$dF = F_x\,dx + F_y\,dy = U_x\,dx + iV_x\,dx + U_y dy + ...
0
votes
2answers
40 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...
0
votes
0answers
36 views

GCD for real analytic functions

In the theory of real analytic functions of several variables, is there a concept of greatest common divisor. If so, does it also hold true that if the gcd of a collection of functions is $1$, then ...
1
vote
3answers
183 views

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg$

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg.$ My main attempt was using the Cauchy-Riemann equations on the product in this manner but this did not work out. My thinking: ...
0
votes
1answer
74 views

Intersection of zero sets of real analytic functions of two variables

The zero set of a real analytic function cannot contain an open set. If we have two distinct real analytic functions of two variables, can they intersect in more than at isolated points? Since the ...
2
votes
0answers
93 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
1
vote
3answers
108 views

Find a harmonic function in the first quadrant,

Find a harmonic function $\phi$(x,y) in the first quadrant with the boundary values $\phi$(x,0) = -1 for x>0, and $\phi$(0,y) = 1 for y>0. Is this function unique? My attempt was this: Consider ...
0
votes
0answers
41 views

Derivatives of the reciprocal of a smooth function

I am trying to find a smooth function f(t) such that its n-th derivatives are bounded by the n-th derivatives of $Ce^{Ct}$, $\forall n \in N$ and the n-th derivatives of its reciprocal are ...
1
vote
1answer
86 views

If all derivatives are zero at a point, what does this imply?

Let's say I have a function $f$ which for all positive $n$ and some complex point $z_0$ satisfies $f^{(n)}(z_0) = 0$. What does this say about the function's analyticity or holomorphicity? Obviously, ...