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
2answers
52 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
47 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
33 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
32 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
85 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
41 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
38 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
52 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
23 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
33 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
35 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
21 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
43 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
33 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
95 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
65 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
77 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
87 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
35 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
60 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, ...
0
votes
2answers
50 views

Nonnegative analytic function as a square

It is know that if $f:\mathbb{C}\to \mathbb{C}^*$ is a continuous function, then for every $n>0$ there exists a continuous function $g:\mathbb{C}\to \mathbb{C}^*$ such that $f=g^n$. Is it true ...
7
votes
1answer
105 views

Does there necessarily exist such a holomorphic function?

This is an old qual problem I'm working on: Let $f:[0,1]\rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Does there necessarily exist a holomorphic function $g: \mathbb{C}\setminus\{0\}\rightarrow ...
4
votes
1answer
148 views

Integrate by parts to prove that this integral provides an analytic continuation ,

Suppose $f(z) = \sum_0^\infty a_nz^n$ converges for $|z| \le 1$. a) Prove $\phi(z) = \sum_0^\infty \frac{a_n}{n!}z^n$ is entire and $|\phi(z)|\le Me^{|z|}$. b) Prove $f(z) = \int_0^\infty ...
2
votes
1answer
85 views

Contour integrals in complex analysis that don't use a closed contour - do we have path independence?

I've noticed that the vast majority of integration problems that I work on in complex analysis are on closed contours, using the Residue Theorem. (If the contour is not closed, we usually close it ...
1
vote
1answer
54 views

Picard's theorem application

I'm trying to solve the following exercise Be $f:\{B\{0,r\} - \{0\}\} \longrightarrow \mathbb{C},\ r>0 $ holomorphic such that in ${0}$ has an essential singularity. Show that if $$f(\{B\{0,r\} ...
0
votes
0answers
80 views

curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider ...
1
vote
0answers
53 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
1
vote
1answer
85 views

Application of Baire Category theorem

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to ...
6
votes
1answer
170 views

Conjecture on zeros of analytic function

I have a conjecture that I can´t prove nor disprove, any help on doing so will be very grateful. Let $f: \{z: |z|<2\} \to \mathbb C$ be a non constant analytic function such that if $|z|=1$ ...
5
votes
1answer
88 views

Identity Principle type question: Prove that $f=g$

While reading a complex analysis textbook the following assertion came up Since $f,g:D\equiv D(a,r) \to \mathbb{C}$ are analytic and injective functions such that $f(D)=g(D)$, $f(a)=g(a)$ and ...
1
vote
0answers
22 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
1
vote
1answer
90 views

Is an analytic one-to-one function on the whole plane necessarily a polynomial? (Can it be disproved?)

I had to show what a one-to-one analytic function from the plane to itself could possibly be. So, I studied the behavior of such a function at infinity: Case 1: Such a function cannot have no ...
0
votes
1answer
79 views

Characterization of analytic functions

First, see this link on the alternative characterizations of analytic functions. I want to prove a version of 3) for complex-analytic functions. In particular: If $f$ is a complex-analytic function ...
5
votes
2answers
121 views

Is this a Morera´s Theorem Application?

Let $G \subset \mathbb C$ be a domain and $f: G \to \mathbb C$ a continuous function such that for any closed and rectifiable path $\gamma \subset G$, $$ \left| \oint_\gamma f(z)dz \right|\leq \left( ...
4
votes
2answers
68 views

What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty ...
3
votes
1answer
44 views

Analytic continuation of a certain Dirichlet series

Is there an elementary way to analytically continue $$f(s)=\sum_{n=1}^\infty \frac{(-1)^n}{(2n+1)^s}$$ to the entire complex plane? It is not hard to see (by grouping terms in pairs and using the ...
1
vote
1answer
31 views

How to prove that a Banach space of analytic functions containing $H^\infty$ except the origin is simply connected?

If $X$ is a Banach space of analytic functions on the unit disk $D$ which contains the space of analytic bounded functions on $D$, how can I prove that $X\setminus\{0\}$ is simply connected?
3
votes
1answer
165 views

Why is this map a Möbius transformation?

Question: Let $D_2=\bar D(2,1)$ and $D_{-2}=\bar D(-2,1)$ be the closed disks of radius $1$ centered at $z=2$ and $z=-2$ in the complex plane, respectively. Set $X= \mathbb C-\{D_2 \cup D_{-2} \}$, ...
6
votes
2answers
101 views

Fundamental solution to the Poisson equation by Fourier transform

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ ...
3
votes
0answers
49 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
2
votes
0answers
23 views

How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
4
votes
1answer
189 views

Are bounded analytic functions on the unit disk continuous on the unit circle?

Let $f(z)$ be holomorphic on the open disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Moreover, let $f$ be bounded on the boundary of $\mathbb{D}$, i.e. $$ \sup_{\varphi \in [0,2\pi]} ...
0
votes
0answers
69 views

Series $1$, $1$, $\frac12$, $\frac12$, $\frac13$, $\frac13$, etc.

Is there any way to define an analytical function in a region that's contained by 0 and 1 that will correspond with the following series: $1$, $1$, $\frac12$, $\frac12$, $\frac13$, $\frac13$, ...
1
vote
0answers
27 views

How to apply Cauchy-Kowalevsky Theorem.

The Cauchy-Kowalevsky theorem is stated in my notes as: For the Cauchy problem: $$ \begin{cases} u_{y}=F(x,y,u,u_{x}) \\ u(x,0)=h(x) \end{cases} $$ If $h$ is analytic in a neighborhood of ...
1
vote
3answers
72 views

Show that f must be constant on C

This is a problem that I have been encountered after reading about analytic functions in complex analysis. Suppose $f(z) = f(x + iy)$ is analytic on $\mathbb{C}$. Let $u= \Re ~f$ and $v = \Im ~f$. ...
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 ...