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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33 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 ...
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31 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 ...
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34 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 ...
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1answer
57 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 ...
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38 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 $$u(x,y)=x^3+xy^2$$ $$v(x,y)=y^3+yx^2$$ $$u_x(x,y) = 3x^2 + y^2$$ ...
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64 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 ...
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19 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 ...
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27 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 ...
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1answer
27 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 ...
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12 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 ...
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47 views

Complex Analysis. Analytic functions

How could I solve this problem?: "Supose an open set A $\subset$ $\mathbb C$ , so that $A^*= \lbrace z \in \mathbb C : \bar{z} \in A \rbrace$. If f is an analytic function in A, demonstrate that ...
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14 views

A multivariable real analytic funtion related to a family of multi-complex-variable analytic functions

Let $D \subset \mathbb{C}^n$ be a open region. Let $f_k(k=1,2,...)$ are holomorphic on $D$ and $u(z):=\sum_{k=1}^{\infty}|f_k(z)|^2$ is uniformly convergent on every compact subset of $D$. Show that ...
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18 views

Degree of zero of a family of real analytic functions on a common interval

Given a family of real analytic functions on a common interval on the real line expanded about a common zero of the family of analytic functions, what can be said about the multiplicity of this zero ...
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21 views

If the result of differentiating a function converges can we claim that there are no singularities in the function?

I was trying to understand an answer to another question of mine Showing Weierstrass Elliptic Function is meromorphic in which the answerer has used "You can differentiate the function term by ...
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37 views

Show that the function $f(z)$ is analytic

Question : If $\phi$ and $\psi$ ae function of $ x $ and $y$ satisfying laplace's equation . Show that $f(z) = s + it$ is analytic , where $$ s = \frac{\partial \phi}{\partial y} - \frac{\partial ...
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19 views

Complex Function With No Singularities

Suppose it is given that a function f is meromorphic (no singularities except poles) and now if in any region it is given that f has no poles also, then can I assume that f is analytic/holomorphic in ...
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34 views

Branch cut for arcsin(z)

I am referring to this particular example found here: http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/arcsin.pdf On page one, I have difficulty understanding the region where $Arcsin(z)$ is ...
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3answers
40 views

An Elliptic function can not be holomorphic/analytic?

I was reading about elliptic functions on the wiki and it said that a doubly periodic meromorphic function in contention of being an elliptic function can not be analytic/holomorphic as it would then ...
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35 views

Can a meromorphic function which does not have any poles in a domain be shown to be bounded in that domain?

Basically such a function would not have any singularities in that domain and is completely analytic on it. Then, is the function bounded on that domain?
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Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...
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34 views

Does analytic continuation apply only to analytic functions?

I'm a high school senior attempting to do a project on the riemann zeta function. I've looked online, tried reading college textbooks but still don't have a completely clear idea of analytic ...
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53 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
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17 views

Is definite integral of such function multiplication analytic?

If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about $$H(s)=\int_{a}^b f(x) g(s,x) dx $$ Is $H(s)$ analytic ...
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38 views

The radius of convergence of a power series about a point interior to the domain of an analytic function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real analytic function with domain an open, non-empty set $(a, b) \subseteq \mathbb{R}$, $-\infty \leq a < b \leq \infty$ and let $c \in (a, b)$. ...
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1answer
57 views

Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$

Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$. Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. ...
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1answer
82 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
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33 views

Are the coefficients of power series expansion for a real analytic function bounded?

Are the coefficients of power series expansion for a real analytic function bounded? $f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n$ We have a sequence $\{a_n\}, n=0,\cdots,\infty$. Is this sequence bounded? ...
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49 views

Analytical solution to nonlinear ode

I solved this equation that I attached numerically in matlab by the Newton Raphson method. Now I want to solve it analytically in matlab or even in Maple if it is possible. Would you please help me ...
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95 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
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40 views

Why is $1/z$ analytic at infinity?

I was given this proof: Let $w(z)=1/z$, so $w$ maps origin to inifinity and infinity to origin. Consider $f(z) = z$. It has no singularities in finite $z$-plane. So $f(w) = 1/w$ has a pole at the ...
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23 views

Analytic branch of root

I'm trying to prove that if $f$ is holomorphic function in an open subset $G$ of the complex plane, and $z_0 \in G$ is a zero of $f$ of order $m$ then there exist a branch of the $m$th root of $f$ - ...
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1answer
24 views

Finding all the analyitical function in the unit annulus that satisfy a given condition for natural numbers

Let $f$ be an analytic function in the annulus $0 < |z| < 1 $ such that it's singularity in $z=0$ is not essential. I want to find all of such functions $f$ that satisfy for $n = 3, 4,...$: ...
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21 views

Showing that $n^z$ for $n\in\mathbb{Z}$ and $z\in\mathbb{C}$ is analytic

I just had a quick question. Can I just say that because we know $n^m$ for any integer $m$ is entire, and since we know that $z$ is entire, then the composition, $n^z$ is entire?
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62 views

A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of ...
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1answer
81 views

If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
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62 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
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1answer
30 views

Holomorphic function is zero on an analytic set then $df=0$.

Assume we have an homomorphic function $f:U\rightarrow \mathbb{C} $ which is holomorphic on the open set $U$ of $\mathbb{C}^n$. Assume there is $V\subset U$ analytic and that $f$ restricted to $V$ ...
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29 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
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1answer
29 views

Determining if a function is real anaytic at the point $a$?

Is there a method, other then using refer to the Taylor series to determine if a real function is analytic at the point $a$. If so please, if possible, could you give a source.
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25 views

Single value function defined in the plane cut

I'm confused about multi/single value complex function. Can someone explain why the function $\sqrt{1-z^2}$ can be thought of as a single valued in the plane cut along $-1\leq z\leq 1$
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44 views

Existence of such function

So we know that if $g(z)=\frac{z-c}{1-\overline{c}z}$ $(c\in\mathbb{C})$ $|g(z)|=1$ for $|z|=1$. Does there exist a function $f(z)$ satisfies the following properties: (1) $f$ is analytic in some ...
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1answer
14 views

Sequence of partial sums converges locally uniformly?

Suppose $f: U \to \mathbb{R}$ is a real analytic function defined by $f(x)=\sum_1^\infty a_n (x-x_0)^n$ and let $f_N=\sum_1^N a_n(x-x_0)^n$. Then $\{f_N\}$ converges to $f$ pointwise. Wikipedia says ...
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70 views

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
2
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1answer
93 views

An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. I am editing my question, since there are duplicates on this forum to the question of why an ...
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2answers
76 views

Is any rational function $R(x)$ a real analytic function in its domain?

To begin with, the definition of a rational function $R(x)$ can be found in Wiki. Suppose that $R(x)$ is defined in a subset $D \subseteq \mathbb{R}^n$. Then my question is: Is any rational ...
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42 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
3
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1answer
71 views

Proving analytic continuation, choosing suitable branch cuts,

Consider the function $$f(z)=\log[(z^2+1)^{1/2}],\quad z>0$$ where the branch is chosen so that $(z^2+1)^{1/2}>0$ for $z>0$ and the log denotes the principal branch. Let $R$ be the union of ...
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1answer
58 views

how to prove that a function is not complex differentiable

I was working on a problem on the complex differentiability of the following function: $f(z)= z \operatorname{Re}(z)$. How to find the points where the given function is not differentiable. My ...
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33 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
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0answers
39 views

Zero of non analytic function

Let a function $L=L(z)$ be analytic, for $\mathrm{Re}\, z>0$, and be singular at $0$, however, $L(0)=c$ be finite. Let also $L'(0)$ be finite as well, however, $L'(0)\neq 0$. For example, $$ ...