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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1answer
170 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
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3answers
31 views

how can I show that $a^n\rightarrow 0$ if $0<a<1$

It is clear for the high school students, but I must use the 'elementary analysis' way to prove the statement. I tried to show it by assuming the contrary, but it didn't give me a good result.
6
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2answers
218 views

Property of sum $\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}$

Is it true that for all $n\in\mathbb{N}$, \begin{align}f(n)=\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}\end{align} is always rational. I have calculated via Mathematica, which says ...
0
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1answer
107 views

How do we find sub-sequences and limit points?

In my lecture notes I'm given the definition of a limit point as: A real number $a$ is called a limit point of a sequence $s_n$ where $n$ is a natural number, if there exists a subsequence $s_{n_k}$ ...
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0answers
51 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
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1answer
27 views

How do i analyze this complex diagram?

I'm asking how to analyze diagrams like this : http://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complex_LogGamma.jpg/600px-Complex_LogGamma.jpg What do distinct colors here mean? What do the ...
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1answer
44 views

Power Series to solve non linear differential equations.

I've been revising Power series recently and their application when it comes to solving linear differential equations, but in this question I'm not sure what to do when it's a non linear function. I ...
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1answer
67 views

Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a ...
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0answers
90 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...
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0answers
30 views

Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
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1answer
86 views

An approach for the analytical continuation of the Gamma Function

I found online these notes, http://math.arizona.edu/~flaschka/COURSES/527/527Notes/Gamma.pdf On the last pages there is a line of reasoning which I am struggling to understand, I repeat herein for ...
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1answer
64 views

Real analysis: Continuity of a function

Define $f: [0,1) \cup [2,3] \rightarrow [0,2]$ by $$f(x)=\begin{cases} x & x \in [0,1) \\ x-1 & x \in [2,3] \end{cases}$$ Is the function continuous at $x=1$? Is the function continuous at ...
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3answers
65 views

Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
7
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1answer
91 views

On derivatives that are not Riemann integrable

Let $f:\;[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$. It is not a mystery that $f'$ need not be Riemann integrable. In fact even if we require $f'$ to be bounded the implication is still false. ...
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1answer
23 views

How can I show that a “function” assigns exactly one value to each x in its specified domain?

I am teaching myself using a book, Mathematical Analysis, authored by Bernd S. W. Schröder. In Exercise 1-30 (b) (i) at page 16, it says that Function $f:Q\rightarrow \tilde{R}$ is described in some ...
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1answer
45 views

Confusion about compact subsets of metric spaces being closed

In Rudin's Analysis, we have Theorem: Compact subsets of metric spaces are closed. Can't I generate a counterexample? $\mathbb R$ is a metric space. $(0,1)\in\mathbb R$ is a subset which is ...
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2answers
70 views

Compact subset of a closed subspace: compact in the whole space?

Imagine that you have a topological space $(X,\tau_{X})$ and a closed subset $Y$. Say that within $Y$ we have a subset $K$ that is compact in the subspace topology $\tau_Y$. Is $K$ compact in ...
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1answer
116 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
1
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1answer
62 views

Fractional part optimization algorithm

I was trying, out of curiosity, to find an efficient algorithm for the problem below which peaked my interest: Let $r$ be a real number. Find an integer $k > 0$ such that $kr$ is "near" an ...
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2answers
35 views

Set of Upper Bounds

Let S be a subset of the reals, and suppose that S is bounded above. Let B be the set of upper bounds of S and suppose that B has no lower bound. What do you conclude about S? I know that it probably ...
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0answers
41 views

Show that a series converges

I'm very new to analysis, so this may appear quite simple. I understand intuitively why, but can't get it down formally. $$\text{Let } {x_n} \text{ be a sequence of real numbers. Suppose } x_n \to ...
3
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1answer
87 views

Derivative and lipschitz

If I have a real-valued continuous function defined on a compact subset of real line, such that its derivative(wherever it exists) is bounded. Is such a function necessarily Lipschitz? Additionally, ...
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0answers
31 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
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1answer
50 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
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0answers
62 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...
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0answers
41 views

Question about Morse index

in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of ...
5
votes
4answers
108 views

Does there exist an $x$ such that $3^x = x^2$?

I tried solving for $x$ by using $x \log(3) = \log(x^2) $$\log(3) = \frac{\log(x^2)}{x}$$ I'm stuck on this part. how do I isolate $x$ by itself? Any help would be appreciated.
0
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1answer
27 views

If $a>0,b>0$ show that exists $x,y\in \mathbb{I}$ near to $a,b$ such that $x^y\in \mathbb{Q}$

If $a>0,b>0$ show that exists $x_0,y_0\in \mathbb{I}$ (irrationals) near to $a,b$ respectively such that $x_0^{y_0} \in \mathbb{Q}$. I was trying this way: Defining the function ...
0
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1answer
60 views

Can we take the infimum over a variable set?

Suppose that we have a family of functions $\lbrace f_{\alpha}(x)\rbrace_{\alpha}$ define on an open set of $\mathbb{R}^{m}$, and $\alpha$ runs over a set $\Gamma$. Assume that the family is ...
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1answer
37 views

Question about relative singular homology groups

I know that the sphere $S^{\infty}$ is contractible, but why if $H$ is a Hilbert space then we have $$H_q(H,S^{\infty})=0, q\in \mathbb{N}?$$ Please help me Thank you
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0answers
31 views

Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
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1answer
36 views

Proof of $\left |\frac{\sin(n+1/2)t}{\sin{t/2}}-\frac{\sin{nt}}{\tan{t/2}}\right| \leq 1$

I need help to proof $$\left |\frac{\sin(n+1/2)t}{\sin{t/2}}-\frac{\sin{nt}}{\tan{t/2}}\right| \leq 1$$
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2answers
101 views

Locality of Inverse Function Theorem

For the Inverse function theorem, the theorem proved the existence of a inverse relation on a local scale, that is if $Df(x)$ is invertible, $f$ is $C^1$ function and $f$: open set E $\subset R^2$ -> ...
1
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1answer
57 views

How to find the Fourier transform of $\mathbf 1_{[0,2\pi]}(x)\sin(x)$?

How does one find the Fourier transform of $f(x):=\mathbf 1_{[0,2\pi]}(x)\sin(x)$? I have tried to use the definition from my text: \begin{align*} \hat f(\xi) & = \frac{1}{\sqrt{2\pi}} ...
2
votes
1answer
60 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
1
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1answer
29 views

Calculating similarity of gaps between integers in a set

First off I should state that I'm not a mathematician, I'm a programmer (Python, Javascript). But I thought this was more of a mathematical question than a programming one, so I'm asking it here. I ...
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2answers
124 views

For which $\alpha$ does the series $\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}}\!\! - \!1\big)$ converge?

For what $\alpha$, does $\displaystyle\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}} - 1\big)$ converge? Divergence of $\sum_{n = 1}^{\infty}(2^\frac{1}{n} - 1)$ prompted this question.
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1answer
39 views

Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent

Let P be a vector space of polynomials with real coefficients. Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent, where $|p|_1$=max$ \{|p(t)|$; $0\leq t \leq 1 \}$ and $|p|_2$ = max ...
2
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1answer
106 views

Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
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1answer
173 views

the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$. Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f ...
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0answers
48 views

Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
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3answers
79 views

Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval?

Is it possible that for two functions $f$ and $g$ and some interval $(a,b)$ we have $f(x)=g(x)$ for all $x\in(a,b)$ and $f(x)\neq g(x)$ for $x$ outside the interval $(a,b)$? $f$ and $g$ are ...
2
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1answer
103 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
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1answer
61 views

proving differentiability of functions in $\mathbb{R}^2$

Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $$f(x,y):= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text{when} \, (x,y) \neq (0,0) \\ 0 &\text{when} \, (x,y)=(0,0)\end{cases}$$ Prove ...
2
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1answer
32 views

Contradiction between $a_0$ and $a_k$ for Fourier Series

I need to calculate the Fourier Series for the function $f(x) = |x| \; f:[-\pi,\pi] \to \mathbb{R}$ When calculating $a_k = {1 \over \pi} \int_{-\pi}^{\pi} f(x) \cos{(kx)} dx \; (k \in \mathbb{N_0})$ ...
2
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1answer
75 views

Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
1
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1answer
93 views

Suprema Proof: $|\sup A-\sup B| \leq \sup|A - B|$

I am trying to prove that $|\sup A(x)-\sup B(x)| \leq \sup|A(x) - B(x)| \quad \forall x$ in some arbitrary set $S$. It is clear why this is true: the difference between the maximum values of each ...
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1answer
77 views

Example of continuous function on a closed unit ball with no minimum point at its sphere

Find a continuous function f from a closed unit ball $\subset R^2$ -> $R^2$ that is continuously differentiable on the unit ball (B(0,1)), but 0 is not in the range of f. and there is no point $X_0$ ...
5
votes
2answers
975 views

Measurable function remaining constant

This is a problem which appeared in one of my tests, which i wasn't able to solve. Let $\Omega$ be a uncountable set. Let $S$ be the collection of subsets of $\Omega$ given by: $A \in S$ if and only ...
1
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1answer
21 views

Suprema and Infima of nonpositive functions

I am trying to get some estimates using the time-dependent infimum and supremum of a function $g(t,x)$. I have the following question. Suppose $g(t,x)\leq0$ for all $x\in\mathbb{R}$ and $t\geq0$. ...