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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185 views

Existence of a measure

I need help in showing that if $\alpha$ and $\beta$ are measures defined on $\mathfrak{A}$, and $\beta \leqslant\alpha$ then there is a measure $\lambda$ on $\mathfrak{A}$ such that ...
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1answer
67 views

How to construct an increasing sequence of sets.

This is part of a bigger problem I'm working on. To construct a a decreasing sequence of sets, $A_{n}\supseteq A_{n+1}$, I did the following: Let $B=\cup_{n=1}^{\infty} B_n$ and set $A_n= B\setminus ...
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3answers
246 views

How many solutions has $z^\pi = 1$?

I know that for $z \in \mathbb C$ and some natural $n\geq 1$, the equation $z^n = 1$ has exactly $n$ solutions. But what if I say $n$ need not be natural, e.g. $$ z^\pi = 1.$$ I mean the equation ...
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4answers
169 views

simple real analysis question on integration

it is trivial that $\int_0^{2\pi} \cos(x)\,dx = 0$. Intuitively, it is clear that for a strictly decreasing positive function $f(x)$, $$ \int_0^{2\pi} f(x) \cos(x)\,dx \ge 0 $$ but I have no glue ...
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2answers
147 views

Maximum in a cell intersected with a sphere

I have a rectangular cuboid-shaped 3D "cell" with scalar values at each vertex $(v_1,\ldots,v_8)$. Within this cuboid I do tri-linear interpolation. What I want is the maximum value of that function ...
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2answers
354 views

Divergent Series

Thinking about divergent series and ways of "summing" them, they seem to fall into two categories (roughly): Series like $\sum_{k=1}^\infty \frac{1}{k}$, which defy all kinds of regularization or ...
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1answer
1k views

Complex Analysis: Laurent Expansion for $\frac{1}{\sin(z)}$ on different annuli.

I've been working through Hilary Priestley's Book Complex Analysis (fantastic read) and have reached her discussion of the Laurent Expansion for holomorphic functions. Considering the function ...
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2answers
186 views

What is the limit of $\lim\limits_{n\rightarrow\infty}\frac{1}{n^4}\left(\sum_{k=1}^{n}\ k^2\int_{k}^{k+1}x\ln\big((x-k)(k+1-x)\big)dx\right)$

As the topic how to find the limit of $$\lim_{n\rightarrow\infty}\frac{1}{n^4}\left(\sum_{k=1}^{n}\ k^2\int_{k}^{k+1}x\ln\big((x-k)(k+1-x)\big)dx\right)\;.$$
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Difficulties with Chapter 2 in Rudin

I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems ...
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2answers
376 views

Analysis on Manifolds Munkres Integration

I needed help in showing that the set $R^{n-1} \times 0$ has measure zero in $R^n$. What I have so far: Let $\epsilon > 0$. If $i_1,\dots,i_{n-1}$ are integers, then define ...
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2answers
106 views

In such way some condition implies nonexistence of derivative?

Assume that a function $f:[0,1] \rightarrow \mathbb{R}$ is continuous. In what way condition $$\forall_{n\in \mathbb{N}} \forall_{0\leq x\leq 1-\frac{1}{n}} \exists_{0<h<1-x} \left| ...
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3answers
229 views

Intermediate growth rates

Is there any simple function/formula $f(n)$, which eventually dominates every $cn$ for every $c$, and is eventually dominated by $a \cdot n \cdot \ln^k(n)$ for every $a,k \in \mathbb{Z}$, where ...
0
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2answers
361 views

Why does the series $\sum_k\frac{(-1)^k}{k+x^2}$ converge uniformly but not absolutely?

Consider the series $\sum_k\frac{(-1)^k}{k+x^2}$. Why does it converge uniformly on $[0,\infty)$? and why doesn't it converge absolutely, always on $[0,\infty)$? The only thing that I noticed is that ...
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4answers
252 views

Borel Measure such that integrating a polynomial yields the derivative at a point

Does there exist a signed regular Borel measure such that $$ \int_0^1 p(x) d\mu(x) = p'(0) $$ for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure ...
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5answers
3k views

Need Help: Any good textbook in undergrad multi-variable analysis/calculus?

This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: -Differentiability. -Open mapping theorem. ...
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591 views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
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3answers
516 views

Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$

In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is ...
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3answers
266 views

How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
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1answer
750 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
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1answer
615 views

An existence of global solution of differential equation of first order

Let $f: (a,b) \times \mathbb{R} \rightarrow \mathbb{R}$ be of class $C^1$ in $D:=(a,b) \times \mathbb{R}$ and satisfies condition $$| f(t,x)| \leq A+B|x| \textrm{ for } (t,x) \in D,$$ where $A,B$ ...
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1answer
187 views

Problem with Picard Iteration

I have $ \frac{dy}{dx} = y^2, y(0) = y_0 $ I have solved this as $y = \frac{y_0}{1 - x y_0}$ Which has the Taylor expansion $ y_0+y_0^2 x+y_0^3 x^2+y_0^4 x^3+y_0^5 x^4+ ...$ However, when I ...
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2answers
287 views

Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$

Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
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2answers
202 views

$f(x,y)$ in polar coordinates

So, I have $ f(x,y) = (x^2-y^2, 2xy) $, which is a local $\mathcal C^1$ isomorphism in $\mathbb R^2 \setminus \{(0,0)\}$. I have to write this function in polar coordinates: $$f(x,y) = ...
8
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3answers
601 views

Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
2
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1answer
207 views

What is this series called and when does it diverge?

What is this series called (if it has a name)? When does it diverge without analytic continuation and when does it diverge with analytic continuation? $\sum_{k_1,\dots,k_m=1}^{\infty} ...
0
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1answer
521 views

Could explain me how eigenvector helps with compute gradient and how make differentiate operation on decrete space like digital image?

Could you explain me how eigenvector helps with compute gradient and how make differentiate operation on descrete space like digital image? I know that this question is so connected with computer ...
2
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0answers
187 views

Solution to polynomial equations with non-radicals

For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial ...
0
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1answer
77 views

finding the function

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which is continuous and satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How ...
2
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1answer
195 views

Requirement of closed and bounded set $[a,b]$ in the Ascoli theorem

In Wikipedia, the Ascoli theorem requires the functions to be continuous on the closed and bounded interval $[a,b]$. However, in the proof given in the book "Theory of Ordinary Differential ...
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0answers
159 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
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1answer
190 views

Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$?

Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $$\tilde f(k) = \frac{\mathcal ...
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1answer
70 views

weak solutions need to have local integrability condition?

I am currently studying Poisson and Laplace equations. This is just a small question that has been causing me some confusion, and I would like some clarification before I resume my study. For example, ...
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2answers
593 views

The primitive of a discontinuous function?

I'm thinking about the primitive of an arbitrary function. As we all know, every continuous function has a primitive according to the Newton-Leibniz formula, $\int_{x_0}^{x} f(x) dx$ is the primitive ...
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0answers
100 views

Simple Swapping Integral Question (A Proof in Univalent Functions by Pommerenke)

This is really a simple question involving swapping the order of integration. The step I'm confused about comes from a proof in Pommerenke's "Univalent Functions," and for those of you with a ...
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4answers
1k views

Every Cauchy sequence in a metric space is bounded

Is the following correct or along the right lines? Thanks for any help Question A sequence $\{x_n\}$ in a metric space is said to be bounded if it is contained in some open ball $B(a,r)$. Prove ...
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3answers
206 views

Why don't we define the class of $C^{\infty}$ in this way?

$C^{\infty}$ is defined to be the class of functions which have all orders of derivative. But as a convention, as far as the infinity is concerned, we always refer to limit. So why don't consider the ...
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2answers
171 views

Does anyone know how to prove this inequality

Does anyone know how to prove the following inequality ...
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0answers
409 views

Taylor theorem for function of several variables

Where one could find the proof of the following version of Taylor theorem for functions of several variables? Assume that $f$ is a function of class $C^{n+k}$ defined in a neighbourhood $W$ of zero ...
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2answers
247 views

Existence of some sort of 'infinite algebraicity' of transcendental numbers

Given an arbitrary number, say, $\alpha \in \mathbb{C}$, can anyone supply either (a) a reason for the existence (or general non-existence) of, or (b) the reverse engineering of a (convergent) ...
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1answer
78 views

Rate of convergence of double sequences

Suppose $ \{X_{n,m} \}$ be a double sequence of real numbers and suppose $\lim_{n}\lim_{m}X_{n,m}=X$. What is the definition and reference for the rate of convergence of double sequences?
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How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
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1k views

Differentiation under the integral sign for Lebesgue integrable derivative

The problem is the following: Let $a,b,c,d \in \mathbb R$ be given such that $a<b$ and $c<d$. Suppose $f: [a,b]\times [c,d] \to \mathbb R$ is a function such that $\partial_1 f: [a,b]\times ...
102
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5answers
7k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
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2answers
574 views

Smoothing a Sobolev function

Let $u \in H^1({\mathbb R}^n)$, $n \geq 2$. Let $\varphi \in C^\infty_0({\mathbb R}^n)$ with $\varphi \geq 0$. Let $\eta$ be a smoothing kernel with $\eta \in C^\infty_0({\mathbb R}^n)$, $\eta \geq ...
5
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4answers
529 views

History of analysis?

Any sites detailing the history of analysis post 1820 (to mid 1900s?) - vis-à-vis Cauchy, Weierstrass, Riemann, Bolzano, ..., Kuratowski, Hilbert? It's something that appears quite interesting and I ...
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1answer
257 views

Theorem 191 from the book Inequalities of Hardy, Littlewood and Pólya.

In the pic, in the second proof of Thm 191 (the one that starts at the paragraph:"We can prove theorem 191 without appealing to the more difficult theorem 190..."), I don't understand why: $$\int ...
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3answers
381 views

Inequality for logarithms

I conjecture the following inequality is true $$\ln x \le (x - 1)\ln\frac{x}{x-1}$$ for all $x > 1$, but I cannot give a proof. I will appreciate if someone can provide one.
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1answer
103 views

How to show $\int_{\mathbb{R}^d} u(x) e^{-|x|^2} (e^{-|x|^2 / n} - 1)^2 dx \rightarrow 0$, if $u\in L^2(\Bbb R^n)$?

How can I prove that $$ \int_{\mathbb{R}^d} u(x) e^{-|x|^2} (e^{-|x|^2 / n} - 1)^2 dx \rightarrow 0 $$ as $n \rightarrow \infty$? Here $u \in L^2(\mathbb{R}^n)$. I'm thinking the dominated ...
2
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1answer
200 views

L'Hospital's Rule for $\infty/\infty$

Arthur Mattuck in his Introduction to Analysis book, pg. 220 says, in order to prove L'Hospital's Rule for $\infty/\infty$ case, Let $L=\lim_{x \to \infty} \frac{f'(x)}{g'(x)}$ and choose $a$ ...
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1answer
281 views

Locally Lipschitz implies Lipschitz under equivalent metrics?

Bonjour. For $i=1,2$ let $X_i$ be a non-empty set and $d_i$ a metric $X_i^2 \to \mathbb{R}$. Suppose $f$ is a locally Lipschitz (*) function $(X_1, d_1) \to (X_2, d_2)$. Question. Do there exist ...