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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How to compute using integration the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around a unit circle?

How to compute the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around the unit circle centered at the origin using the methods of the integral calculus?
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12 views

Does the map $F(x, y)=(f(x, y), y)$ induce an isomorphism $dF_{(a, b)}$?

Suppose $f:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ be a $C^p$ map such that $df_{(a, b)}:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ is surjective for some $(a, b)\in\mathbb R^{n+m}$. Define ...
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11 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
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1answer
29 views

What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
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2answers
62 views

Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$
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1answer
26 views

Analysis Arithmetic series .Verify which of the following sequences converge.

Verify which of the following sequences converge.$$1.A(n)=\sum_{n=1,n=+00}(1/(n^{1+1/n})$$ $$2. B(n)=(1/\sqrt{n^2+1})+.......n/\sqrt{n^2+n}$$ $$3.C(n)=(n+cos(n^2))/(n+sin(n)) $$ .For the 3th one ...
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2answers
16 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
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0answers
21 views

Local inversion theorem (théorème d'inversion local)

I don't understand how to use the local inversion theorem to prove that a nondegerate critical point of a function $f\in C^2(U,\mathbb{R})$ is isolated Thank you.
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0answers
43 views

Measure theory problem from Stein real analysis

Let $\mu$ be a Borel measure on the sphere $S^{d-1}$ which is rotation-invariant in the sense that $\mu(r(E)) = \mu(E),$ for every $r$ of $\mathbb{R}^d$ and each Borel subset $E$ of $S^{d-1}.$ If ...
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1answer
29 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
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0answers
18 views

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
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3answers
48 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
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1answer
73 views

Transforming ODEs into exact equations.

I want some examples of ODEs that can only be solved by transforming them into exact equations. They shouldn't be solvable with; Direct integration, separation of variables, manipulating a reverse ...
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2answers
60 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C([0,1])$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} |f(x)-f(x_0)| \leq n|x-x_0| \end{align*} for all $x \in [0,1]$. Why is ...
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1answer
41 views

IMPROPER INTEGRAL of euler formula [on hold]

Find the value of this improper integral given by $\int_\mathbb{R} e^{i \theta} d \theta$.
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4answers
64 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
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1answer
36 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
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1answer
41 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
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1answer
55 views

How to arrive at desired equality?

Why is the following second equality true? $$e^{1+1/2+...+1/(n+1)} - e^{1+1/2+...+1/n} \\= ...
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0answers
40 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
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1answer
33 views

If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?

I was working my way through the set theory chapter in my Analysis textbook when I stumbled across these two theorems: Every infinite set has a countable subset A union of countable subsets ...
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1answer
19 views

Lemniscate curve parametrization exercise

Let $\gamma (t) : \mathbb R \to \mathbb R^2$ be the function $$\gamma(t)=\left(\frac{(1+t^2)t}{1+t^4},\frac{(1-t^2)t}{1+t^4}\right)$$ Prove that the function is $\gamma$ is differentiable, regular ...
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1answer
35 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
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1answer
32 views

Is there an explicit polynomial form for the product of consecutive integers?

I have the product $\prod_{j=0}^{r-1} (n+j)= n(n+1)\cdots(n+r-1)$ where n is a positive integer, and I was wondering if there was an explicit polynomial form for it (as a polynomial of degree r). I've ...
2
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0answers
47 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
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5answers
85 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
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1answer
120 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
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0answers
13 views

Coordinate Change Operator

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be analytic. Recall that for $ h \in \mathbb{R} $, the translated function $ \tilde{f} (x) = f(x+h) $ can be formally written as $ \tilde{f} = e^{ h ...
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1answer
18 views

About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {$ {x∈X∣lim f_{n}∈R}$} $ $ = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞ ${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)}$ $ and that ...
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1answer
16 views

Is this a B-measurable funtion?

Is the function defined by: $f(x)=e^x $ if x is in E and $f(x)=e^{-x} $ if x is not in E measurable?, here f goes from R to R, and E is not member of the Borel-algebra
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0answers
31 views

Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
2
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2answers
49 views

Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
1
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1answer
36 views

Prove that $f(x) > g(x)$ where both functions are convex and have the same value and slope at $0$

Let $f: [-a,a] \to \mathbb{R}$ and $g: [-a,a] \to \mathbb{R}$ be two non-negative, convex and smooth functions. We further know $f(0) = g(0)=0$ and $f'(0) = g'(0)=0$. I'd like to show $$f(x) \ge ...
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2answers
56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
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2answers
49 views

How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...
2
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1answer
69 views

Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

For each real number $x$, let $[x]$ be the largest integer less than or equal to $x$. For example, $$[5] = 5$$ $$[7.9] = 7,$$ and $$[−2.4] = −3.$$ An arithmetic progression of length $k$ is a ...
3
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1answer
65 views

Functions for which $\mathcal{F}g = f \ast f$

Suppose one is given $f \in L^{2}(\mathbb{R})$, my question is whether or not there exists a $g \in L^{1}(\mathbb{R})$ such that $f \ast f = \mathcal{F}g$ where $\mathcal{F}$ is the Fourier transform. ...
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2answers
28 views

How to show $\textrm{supp}(f*g)\subseteq \textrm{supp}(f)+\textrm{supp}(g)$?

Let $f, g\in C_0(\mathbb R^n)$ where $C_0(\mathbb R^n)$ is the set of all continuous functions on $\mathbb R^n$ with compact support. In this case $$(f*g)(x)=\int_{\mathbb R^n} f(x-y)g(y)\ dy,$$ is ...
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1answer
37 views

Prove limit of function

Let $f:(a,+\infty) \to \mathbb{R}$ and on every finite $(a,b)$ interval function $f$ is bounded. Then $$\lim_{x \to \infty}\frac{f(x)}{x}=\lim_{x \to \infty}f(x+1)-f(x)$$ How can we prove or ...
7
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1answer
92 views

Nonstandard complex numbers and categoricity

Let ${}^*\mathbb{C}$ be a nonstandard complex number field (given, for instance, as a countable ultrapower.) By the transfer principle ${}^*\mathbb{C}$ is algebraically closed of characteristic zero, ...
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2answers
54 views

How should I calculate $\displaystyle\int_{-\infty}^\infty\exp\left\{-\frac{1}{2}(x-it)^2\right\}dx$?

I've read that the residue theorem would help to calculate $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\exp\left\{-\frac{1}{2}(x-it)^2\right\}}_{=:f(x)}dx$$ Since $f$ is an entire function ...
2
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2answers
100 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
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0answers
52 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
1
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1answer
33 views

Entire functions such that $\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty$

The problem I am working on is to find all entire functions satisfying $|f(z)| > 0$ for $|z|$ large and $$\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty.$$ My guess is that ...
3
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0answers
51 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
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0answers
17 views

Is $d\mu = g\, dm$ a Radon measure for $g \in L^{1}(dm)$?

Let $m$ is the standard Lebesgue measure and let $g \in L^{1}(dm)$. We know that $m$ is a Radon measure. Is $\mu$ defined by $d\mu = g\, dm$ also a Radon measure? We first claim that $\mu(K) < ...
0
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1answer
31 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
1
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1answer
11 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
1
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1answer
21 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
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0answers
32 views

Recommendation of analysis textbooks

I'm studying real&complex analysis, functional analysis, measure theory, and Lebesgue integral. I want to make sure which books to use in order to study these fields most rigorously and deeply. ...