Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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241 views

Weak* compactness of the unit ball

Things that we know: In any topological space compactness implies sequential compactness If E is any topological space the then the closed unit ball $$ B_E=\{f\in E^*; \|f\|\leq 1\} $$ is compact ...
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2answers
85 views

What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
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1answer
46 views

A set that is a countable intersection of open and dense sets but not open.

We know that Baire Category Theorem implies that in a complete metric space, the countable intersection of open and dense sets is nonempty and actually dense itself. But it is clear that a countable ...
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1answer
190 views

Reference requestion: Existence/construction of bump functions

I'm not much of an analyst myself, but I've time and time again come across proofs which require knowledge of the existence of bump functions. However, I've never studied them, so I'm missing ...
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2answers
220 views

3rd Order Taylor expansion of $e^x\cos(y)\sin(z)$

I'm looking for the 3rd.-order Taylor approximation of $(x,y,z) \mapsto e^x\cos(y)\sin(z)$ at $(x_0,y_0,z_0) = (0,0,0)$ I've got this piece of advice at hand: $\quad\textit{Use the Taylor series ...
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1answer
465 views

Proof on the existence of n-th roots of non-negative real numbers

I can't figure out one part of the proof of this theorem from this document Proof. It's the part where we suppose $y^{n} < x$ Then we can choose a real number h such that $0 < h < ...
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1answer
60 views

To expand a function to power series

For $$ f(x)=\frac{d}{dx}\left(\dfrac{e^x-1}{x}\right) $$ Where is the convergence zone? How to calculate the series $ \sum_{n=1}^{\infty}\frac{n}{(n+1)!} $?
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1answer
2k views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
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1answer
40 views

Basic question on n-dimensional derivative

Let $f : \mathbb R^n \rightarrow \mathbb R$ , suppose that $f \in C^{2}(\mathbb R^n)$ and suppose too $\inf_{\mathbb R^n} |\nabla f | \geq \alpha > 0$ for some $\alpha$. My intuition says this $$ ...
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1answer
52 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
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0answers
56 views

Compactness hypothesis in Riesz representation theorem

Let $X$ a compact metric space; I have to identify the dual of the set of continuous functions on $X$, $C(X)^*$. By Riesz representation theorem we have that it can be identified with the space of ...
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1answer
50 views

Extension of a holomorphic function in the disc

let $f$ be a continuos function in ${0<|z| \leq r} $ holomorphic in the inner and such that $f(z) $ is real for $|z|=r$. Prove that exist a function $g$ on $\mathbb{C} ^*$ such that $f(z) =g(z) $ ...
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1answer
33 views

Remains true for f integrable?

$f(x)=f(x+1)\ \forall x\Rightarrow \int_0^1f(x+t)dt=\int_0^1f(t)dt$, when $f$ is continuous on $[0,1]$. The proof it is not hard. My question is, this property remains true if $f$ is only integrable? ...
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27 views

Solving two equations for $p$ and $q$

How we obtain from the equations $z^2-pqxy=0$ and $z-a(xp+yq)=0$ that $p=\frac{z}{cx}$ and $q=\frac{cz}{y}$, where $a(c+\frac{1}{c})=1$?
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3answers
47 views

Methods to distinguish continuous probability distributions

I read in the Wikipedia article for Variance The variance is one of several descriptors of a probability distribution. In particular, the variance is one of the moments of a distribution. In that ...
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1answer
54 views

Show that there is no function $\phi\in C^2(\mathbb R^3,\mathbb R)$ such that $\nabla \phi=(-y,x,0)^t$

I want to show that there exists no function $\phi\in C^2(\mathbb R^3,\mathbb R)$ such that $\nabla \phi=(-y,x,0)^t$. I did it this way: I know $\nabla \phi=(\phi_x,\phi_y,\phi_z)^t$. Integrating ...
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1answer
435 views

Jacobian matrix and determinant - relation to orientation

$F$ is a function from $V$ to $V$ where $V$ is a $n$-dimensional vetor space and $p \in V$. In the article Jacobian determinant it says: "If the Jacobian determinant at $p$ is positive, then $F$ ...
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1answer
28 views

Linear functional vs. map

A few days ago we were briefly discussing Taylor's theorem in higher dimensions in the lecture. Referring to the expression $f(x)=f(a)+Df(a)(x-a)+$higher order the lecturer said that in general $Df$ ...
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1answer
43 views

Minimum value of the inequality

Give $a_{1}\geq a_{2}\geq ...\geq a_{n}> 0$ and a positive integer m . Find the minimum value of the following the inequality: $\left ( a_{1}+a_{2}+...+a_{n} \right )\left ( \frac{1}{a_{1}^{m}}+ ...
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3answers
71 views

Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$

Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...
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2answers
71 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
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0answers
30 views

$f(z) = u(x,y) + i\cdot v(x,y)$ holomorphic in a connected open set $D$, such that $a\cdot u(x,y)+b\cdot v(x,y)=c$, is constant

Let $f(z) = u(x,y) + i\cdot v(x,y)$ be a holomorphic function in a connected open set $D$. If $a\cdot u(x,y)+b\cdot v(x,y)=c$ in $D$, where $a,b,c$ are real constants which are not all zero, why ...
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3answers
178 views

When to use $\times$ and $\otimes$

Im wondering when to use $\times$ and when to use $\otimes$. In some cases it seems very straightforward, for example $\times$ can be used when combining two elements into an n-tupel (for a product ...
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1answer
49 views

A continuos and holomorphic function on $D^2$ that take pure imaginary values on $S^1$ is costant

Let $D := \{ |z| < 1\}$ and $f : \overline{D} \rightarrow \mathbb{C}$ be a continuos and holomorphic function on $D$ that take pure imaginary values on $\partial D$. Why $f$ is constant? From ...
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1answer
97 views

Translate a vector field

Imagine that you have a vector field $A = \frac{A_0}{r} e_{\theta}$ in cylindrical coordinates, where $A_0 \in \mathbb{R}$. Now you translate your coordinate system in $e_x$ direction by $x \mapsto x ...
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2answers
112 views

Supremum of two unequal sets are equal

I am reading the first chapter of Principles of Mathematical Analysis by Walter Rudin. On page 4, Example 1.9 says as follows. Let $E_1$ be the set of all $r \in \mathbb{Q}$ with $r < 0$. Let ...
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2answers
215 views

How to prove this series $\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$ diverges

Question: Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent? My idea: since ...
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0answers
38 views

Missing Step in a Paper of Struwe.

In this paper on Page 4, in the last line of the proof, the author asserts that if a radial function $u:\mathbb{R_t}\times \mathbb{R}^2\to \mathbb{R}$, smooth outside the origin $(0_t,0_x)$, admits ...
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0answers
97 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
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1answer
168 views

Question about Hatcher's book CW complex

I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood ...
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1answer
109 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
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2answers
66 views

if $f(x) \sim g(x)$ is $ \sum f(k) \sim \sum g(k)$

if $f(x) \sim g(x)$ as $x \to \infty$ then is $\sum_{k=1}^N f(k) \sim \sum_{k=1}^N g(k)$ as $N \to \infty$? Intuitively, i should think so because as $k$ gets larger $f$ and $g$ get closer so it ...
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3answers
76 views

Rigorous method for showing this limit

Prove the following limit; $$\lim_{x \to +\infty}\dfrac{\exp(x^2)}{10^{|x|}}$$ The limit of this is $=\infty$ But what is the best method to show this: L'Hospital doesn't seem very helpful ...
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0answers
62 views

Abel's Functional Equation for $L(x) = \sum x^{n}/n^{2}$

In "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" Hardy talks of Abel's Functional Equation $$L(x) + L(y) + L(xy) + L\left(\frac{x(1 - y)}{1 - xy}\right) + L\left(\frac{y(1 - ...
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1answer
40 views

Regularity of Boundaries

From my understanding, regularity of boundaries are effectively talking about the continuity of the boundary of a set. For example, if I consider $\Omega \in \mathbb{R}^2$, where $\Omega$ is the unit ...
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2answers
49 views

How to find all the positive integer number $n$ such that $\sum_{i=1}^{n}a_{i}=0$ and $|a_{i}|=1$ has a solution in $\mathbb{C}$ with $a_i+a_j\neq 0$

Find all the $n$ for which there exist complex numbers $a_{1},a_{2},\cdots,a_{n}$ such that: (1):$$|a_{i}|=1,i=1,2,\cdots,n$$ (2):$$a_{1}+a_{2}+\cdots+a_{n}=0$$ (3): for any $i\neq ...
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0answers
102 views

Winding number and homotopy

Given two maps $f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them. Well, we know that we can write them as $f(\exp(2 \pi i ...
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1answer
60 views

An open map from $\mathbb{C} \rightarrow \mathbb{C}$ has open real and imaginary part?

If $f(z) :\mathbb{C} \rightarrow \mathbb{C}$ is an open map such that $f(z) = f_1(z) + if_2(z)$ where $f_1$ and $f_2$ represent respectively his real and imaginary part, we could say that both $f_1$ ...
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1answer
666 views

convolution of function with itself 4 times

I have to compute the convolution of $ f(t) = \frac{1}{\pi}\frac{1}{t^2 + 1} $ with itself 4 times, i.e. $$ f \star f \star f \star f $$ I slightly doubt that doing it in steps, i.e. taking $f \star ...
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1answer
78 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
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3answers
197 views

Question on how to find this particular limit..

Find the following limit $$\lim_{x \to 0^-}\left(\dfrac{\cos(x)}{\sin(x)}-\dfrac{1}{x}\right)$$ I checked online and the correct answer is $-\infty$ but I am not sure how to get to it via ...
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1answer
56 views

CW complex topology

I am looking at the real projective plane and I am supposed to show that is possesses the structure $\mathbb{R}P^n = e_0 \cup\cdots\cup e_n$. Well, I know that $\mathbb{R}P^n = S^n/(x \sim -x)$ I ...
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0answers
56 views

Interchanging index of summation in $d$ dimensions

Let $\alpha = (\alpha_{1}, \ldots, \alpha_{d}) \in \mathbb{Z}_{\geq 0}^{d}$ and let $|\alpha| = \alpha_{1} + \cdots + \alpha_{d}$. I have the following question about interchanging summations: Is ...
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1answer
71 views

Recurrence theorem (Poincaré)

A function $g\colon M\to M$ is called to conserve the volume if for each Jordan-measurable subset $J\subset M$ it is $\text{vol}(g^{-1}(J))=\text{vol}(J)$. Resurrection Theorem by Poincaré: ...
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0answers
54 views

Q: An infinite subset $E$ of a compact set $K$ has a limit point in $K$

I'm having difficulty following this proof and was hoping someone could help give a clear picture of what Rudin is doing. pf If no point of $K$ were a limit point of $E$, then each $q \in K$ would ...
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1answer
17 views

$X$ $T_4$, then $X/R$ $T_4$?

I found out that if $X$ is $T_4$ and $R$ is a closed equivalence relation, then $X/R$ is also $T_4$. I was just wondering whether the same is true for $R$ being an open equivalence relation?
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1answer
50 views

partial derivative of function with a matrix

Let $A$ be a $n\times n$ matrix. Let $f\in C^1(\mathbb R^n)$ and $g:\mathbb R^n\rightarrow\mathbb R, g(x)=f(Ax)$. What is the partial derivative $\partial_{x_i} g(x)$? So $Ax=(\sum_{l=1}^n ...
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4answers
116 views

How do I show that $\lim_{x \rightarrow \infty}(1+\frac{a}{x}+\frac{b}{x^{3/2}})^x =e^{a}$?

How do I show that $\lim_{x \rightarrow \infty}(1+\frac{a}{x}+\frac{b}{x^{3/2}})^x = e^a$? Actually, I had to deal with something similar yesterday and after thinking about it for quite a while I did ...
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1answer
148 views

Hardy-Littlewood maximal theorem (Marcinkiewicz)

I have two pages from a book called "Garnett" and I will present Hardy-Littlewood maximal theorem in class on Wednessday. The theorem is stated: if $f\in L^p(\mathbb{R}), 1 \leq p \leq \infty,$ then ...
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1answer
53 views

$V$ is open , then $V=\{x\in \mathbb R:f(x)>0\}$ for some continuous function $f$

Let $V$ be a non-empty open set of real numbers , then how to prove that there is a continuous function $f:\mathbb R\to \mathbb R$ such that $V=\{x\in \mathbb R:f(x)>0\}$