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

4
votes
0answers
140 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
0
votes
1answer
59 views

Not able to understand a paragraph in John Conway's Complex analysis book.

On page 97 under the heading "Counting zeroes; the open mapping theorem" there is a second paragraph which goes like this: In section 3 it was shown that if an analytic function $f$ had a zero ...
0
votes
1answer
58 views

Prove that $\,\displaystyle f(z) = \sum_{n\ge1}\frac{z^n}{n^2}$ is univalent in the disk $\,D\big(\frac23\big)$

I'm having some difficulty with this question: Prove that the function $\,\,\displaystyle f(z) = \sum_{n=1}^\infty\frac{z^n}{n^2}\,$ is univalent in the disk $D\big(\frac23\big)$. There is the ...
1
vote
2answers
66 views

smooth vs. analytic in the definition of almost-complex manifolds

Let $A_{\infty}\hspace{-0.03 in}$ be a maximal $C^{\infty}\hspace{-0.02 in}$ atlas on $M\hspace{-0.03 in}$, and with that smooth structure on $M$, suppose $\: j : TM\to TM\:$ is a smooth function ...
0
votes
0answers
70 views

Let $f$ be an entire function $\mathbb C$, and $g(z) = \overline { f(\overline z)}$. Which of the following statements are valid? [duplicate]

Let $f$ be an entire function, $\mathbb{C}$, and $g(z) = \overline{f(\overline{z})}$. Which of the following statements are valid? Let $f(z) \in \mathbb{R}, \forall z \in \mathbb{R}$, then $f = g$. ...
0
votes
0answers
80 views

Measure of inverse image of points by an analytic mapping

How can one prove the following statement: For any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of $M$ ...
1
vote
0answers
74 views

Schwarz Reflection Principle for the four quadrants in the plane and for two intersecting circles,

I'm looking at an old exam problem that shows a picture of what the function f does to the plane. On the upper right quadrant, there is a + sign, which indicates that f maps this quadrant one-to-one ...
0
votes
0answers
51 views

Understanding the Schwarz Reflection Principle,

The way I understand the principle is that we look at F, piecewise-defined as: F= $$ z \mapsto f(z), \ z \in \Omega^+$$ $$ z \mapsto \overline{f(\bar z)}, \ z \in \Omega^-$$ Here $\Omega^+$ and ...
0
votes
1answer
52 views

Show that the solution of this differential equation is analytic

Let $\alpha,\beta,a,b$ be real constants. Show that the differential equation given by: $y''= ay' + by \\ y(0)=\alpha\\ y'(0)=\beta$ has an unique solution and this solution is analytic in ...
1
vote
0answers
103 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
0
votes
0answers
29 views

Proof that real power series is real analytic

I'm wondering if the following argument is correct. The proof in the book is longer and I don't understand it. Theorem. Suppose $f(x) = \sum_{n=0}^\infty a_n x^n$, where the series converges for $-R ...
1
vote
0answers
20 views

Analytic structures on $S^1$|

I am currently studying Haefliger's paper "Homotopy and Integrablity". During the last chapter, he applies his theory of $\Gamma$-structures to analytic codimension $1$ foliations. Throughout the ...
1
vote
2answers
35 views

What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$?

let $f:D=\{z\in \mathbb C:|z|<1\} \to \overline D$ with $f(0)=0$ be a holomorphic function. What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$? My try:By cauchy integral formula : ...
2
votes
1answer
80 views

Analyticity of log f(z)

In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is ...
2
votes
0answers
33 views

Application of Bernstein's theorem

There is a theorem due to Bernstein related to analytic functions : If $f : ]0,1[ \to \mathbb{R}$ is an absolutely monotonous function (that is a $\mathcal{C}^\infty$ function such that for ...
0
votes
0answers
19 views

Question about inverse of CDF being a real analytic function

Let F: [0,a] -> [0,1] be a continuous, strictly increasing CDF. Assume also F admists a continuous, positive pdf f. Now define the inverse function h(x) as F(h(x))=x. Is h real analytic? If not, what ...
1
vote
3answers
91 views

Analyticity of $\dfrac{1}{z}$ vs. $\dfrac{1}{z^2}$

I am learning complex analysis on my own. I am familiar with the theorems, and I am able to compute by hand and get correct results. But there is something that escapes me. What is the criteria for ...
-1
votes
1answer
14 views

A Question from complex variable [closed]

Show that an analytic function with constant modulus is itself a constant
1
vote
2answers
71 views

Analytic Functions, Cauchys Integral Formula

Let $f: \mathbb D \to \mathbb D$ be analytic or holomorphic with $f(0)=\frac{1}{2}$ and $f(\frac{1}{2}) = 0$ where $\mathbb {D} = \{ z: |z| \leq 1\}$. Then find $|f^{'}(0)|$ and ...
1
vote
1answer
61 views

Find the residue of the function $g(z)=f(z^2)$ at a given point.

Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...
0
votes
1answer
207 views

Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
0
votes
2answers
86 views

Why does the ring of entire functions have no zero divisors?

Why does the ring of entire functions have no zero divisors, while the ring of infinitely differentiable functions on the real line does?
0
votes
1answer
71 views

Rouche Theorem and applications

Exercise 3. 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) ...
1
vote
0answers
17 views

About smooth function whose reciprocal is also smooth

I know that for exponential functions $e^{at}$, or functions like $1/t^m$, t>0, both they and their reciprocals are smooth. Could you please give me more classes of smooth functions, or Analytic class ...
1
vote
1answer
57 views

How to find some $C^\infty$ functions that do not satisfy the uniqueness theorem for analytic functions

The uniqueness theorem for analytic functions states that suppose two series $\sum_{n=0}^\infty s_nx^n$ and$\sum_{n=0}^\infty t_nx^n$ converges in the interval $(-R,R)$. If the set of $x$ that ...
4
votes
2answers
57 views

$f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?

$f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$,but this ...
1
vote
1answer
110 views

Prove that there is no function $f$ that is analytic. [duplicate]

Prove that there is no function $f$ that is analytic in $\mathbb{C}\setminus\{0\}$ and satisfies $$|f(z)|\geq\frac{1}{\sqrt{|z|}},\quad \operatorname{for all}\quad z\in\mathbb{C}\setminus\{0\}$$ I am ...
1
vote
2answers
50 views

How to show real analyticity without extending to complex plane

Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$ For example, $$f(x)=(1+x^2)^{-1}.$$ Using complex analysis, we can easily show $f$ is real analytic. Is there an easy, general method ...
-1
votes
1answer
38 views

Solving a logarithmic equation with variables on each side

Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. ...
6
votes
2answers
456 views

Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: ...
0
votes
2answers
51 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
0
votes
1answer
62 views

An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I ...
0
votes
1answer
158 views

Are the integrals of the following function path independent in the following domain?

Are the integrals of the function: $$f(z)=\frac{1}{z+1}+\frac{1}{(z+1)^2}+e^{\frac{1}{z}}$$ path independent in the following domain: $$D= \{Re z >0\}\setminus\{1\}$$ My thoughts on the ...
0
votes
1answer
37 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
0
votes
1answer
106 views

Bound on the sup norm for derivatives of a particular $C^\infty$ function

I'm reading textbook "A Primer of Real Analytic Functions" and on page 86 the following "obvious" claim is made: Let $|| \cdot ||$ be the sup norm on $[0, 2 \pi]$ and define function $f$ to be ...
0
votes
1answer
28 views

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. Then, (a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle; (b) the set {z : ...
2
votes
1answer
40 views

Prove $f$ analytic on $D(z_0;R)\setminus\{z_0\}$ implies $\exists M, f(D(z_0;r)\setminus\{z_0\})\supset\{z\in\mathbb{C}:|z|>M\}$

Suppose $f$ is analytic on $D(z_0;R)\setminus\{z_0\}$, and $z_0$ is a pole of $f$. Prove that for any $r\in(0,R)$, there is $M\in(0,\infty)$ such that ...
0
votes
1answer
109 views

Analytic continuation of function differentiable on real line to complex plane

If $f(z)=g(z)$ on $(0, \infty)$ and f(z) is holomorphic on an open set $U \subset \mathbf{C}$ with $(0, \infty) \subset U$, but we do not have any information about where $g(z)$ is holomorphic, can we ...
0
votes
1answer
68 views

derivatives of non-analytic smooth functions

I would like to know how to calculate the derivative of a non-analytic smooth function? Suppose $f:\mathbb R\rightarrow \mathbb R$ is in $\mathcal C^\infty\backslash \mathcal C^\omega$ and in ...
1
vote
1answer
81 views

Analytic continuation of function continuous on boundary

Suppose one has a function $f$ in the disc algebra ie: $f$ is continuous on $|z|\leq1$ and holomorphic in $|z|<1$. I wondered, can $f$ always be extended to a holomorphic function on some region ...
1
vote
2answers
87 views

Let f be an analytic function in the disk … Prove that

Let $f$ be a function analytic in the disk $D={z:|z-1-i| \leq 4}$ that satisfies $|f(z)| \leq 1$ for all $z$ in $D$. Prove that $|f^{(3)} (1-i)| \leq \frac{3}{4}$. Hint: apply Cauchy's Integral ...
1
vote
1answer
52 views

Residue theorem application [demonstration]

I really don't know how to solve this problem! Consider $F$, an analytic fuction, so that, $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial So, I tried to ...
2
votes
1answer
110 views

Holomorphic function on the open unit disc

Question: Let f be a holomorphic function on the unit disc $\{|z|<1\}$, which of the following is/are necessarily true? If for each positive integer n we have $f(1/n)=1/n^2$ then $f(z)=z^2$ on ...
2
votes
1answer
147 views

Number of zeros of a periodic function

Let's consider a periodic real function of a real variable $f(x)$. If the function is analytical and it is not the zero function, can one infer that the number of zeros in one period $[x,x+P)$ is ...
1
vote
0answers
49 views

Finding region where $f(z)=z^2\bar{z}$ is analytic.

How can I find a region where $f(z)=z^2\bar{z}$ is analytic ? I first let $z=x+iy$ ,then use Cauchy-Riemann equation and obtain $$u(x,y)=x^3+xy^2$$ $$v(x,y)=y^3+yx^2$$ $$u_x(x,y) = 3x^2 + y^2$$ ...
3
votes
2answers
69 views

$\frac{1}{z^2}$ is holomorphic

I have to show that $z\mapsto\frac1{z^2}$ is holomorpic on $\mathbb C\setminus\{0\}$ and compute its $n$-th derivative I know that $\frac{1}{z^2}=\sum\limits_{n\ge0}(-1)^n(n+1)(z-1)^n$, so it ...
1
vote
2answers
69 views

Composition of real-analytic functions is real-analytic

Suppose $f,g: \mathbb{R} \to \mathbb{R}$ are real analytic, i.e, locally given by convergent power series. Then $g \circ f$ is real-analytic as well. How do I prove this? I guess the "standard" proof ...
4
votes
2answers
36 views

Properties of holomorphic functions (demonstration)

I don't know how to do this demonstration: "If f is an holomorphic function, and M $\in \mathbb{R}^+$, such that for $z \in \mathbb{C}$, $|f(z)| \leq M(1+ |z|^n)$, then f is a $n$ or less degree ...
1
vote
1answer
46 views

Prove that an entire complex-valued $f$ is constant.

If a complex-valued function $f = u + iv$ is entire with $uv = 3$ for all $z \in \mathbb C$, then $f$ is constant. $f$ is not constant $\rightarrow f^2 = (u^2 - v^2) + 2iuv$ is not constant. Since ...
2
votes
0answers
29 views

What are some general strategies to build measure preserving real-analytic diffeomorphisms?

One could prove the following theorem in the smooth setting: Theorem Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be ...