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
74 views

Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
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0answers
42 views

If a normal subgroup, N, contains a lattice why does G/N have finite measure?

Suppose $G$ is a locally compact Hausdorff topological group and suppose $H \leq N \leq G$ are closed subgroups with $N$ normal. Now suppose $G/H$ has a finite $G$-invariant Boreal measure (in the ...
3
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1answer
41 views

How prove this $f(x)=\sum_{n=1}^{\infty}\left(\frac{\sin \frac{1}{x-r_n} }{2}\right)^n$ such follow condition?

Question: let $r_{1},r_{2},\cdots.$ is the interval $[0, 1]$ rational sequence in a line, and define: $$f(x)=\begin{cases} ...
1
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1answer
55 views

Question about convergence of a power series and when the series is not zero

Following is a past exam question I am trying to solve as a preparation for my own exam. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $a_n \leq M$ for some $M\in\mathbb{R}^{+}$ and ...
17
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5answers
517 views

Proving a certain map on the closed unit disc must be the identity

Bounty expired. Will gladly re-create one if a satisfactory answer is posted in the future. Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the ...
3
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0answers
35 views

Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
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0answers
31 views

Just a basic limit of a function [duplicate]

Find this limit,and if you can, do it without derivates or integrals or l'Hospital , just using the fundamental limits of functions. $$\lim_{x\to1} \left( \frac{x}{x-1} - \frac{1}{\ln(x)} \right)$$
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1answer
73 views

A differentiable function whose derivative is not elementary.

Do we know of any differentiable function whose derivative is not an elementary function? This may be a silly question, but in the light of this answer, as pointed in the comments, finding an example ...
2
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1answer
54 views

How to rigorously prove that these two sets have different order types?

Let $A$ and $B$ be two given ordered sets with the linear (or total) order relations $<_A$ and $<_B$, respectively. Then $(A,<_A)$ and $(B,<_B)$ are said to be of the same type if there ...
2
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2answers
57 views

Derivative: a special tangent

I've learned in Euclidean Geometry that the tangent is a line which pass through only a point. For example, if someone ask me to find the tangent at this point $A$, I can easily say that the tangent ...
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3answers
113 views

Proving $x^{1/5}$ is differentiable

The complete question is to prove that $f: \mathbb{R} \to \mathbb{R}$ $f(x) = x^{1/5}$ is continuous everywhere and differentiable everywhere but at x = 0 so I figured I'd prove that it's ...
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2answers
29 views

Applying the substitution rule for integrals

While I was doing the exercises on the whole I came across this kind of exercises:$$\int \frac{mx+n}{ax+b} \mathrm{d}x $$, my book, I also wrote the execution of my book that is: $$\int ...
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0answers
24 views

Show that the Kelvin transform is bounded

I start with definitions: Consider $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. The function $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ with $$ ...
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1answer
57 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
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1answer
15 views

U does not fade away more weakly than $\frac{1}{\lVert x\rVert^{n-2}}$

Hello I have a function $u$ and it is said that for $n>2$ it does not fade away (in German: abklingen) more weakly than $\frac{1}{\lVert x\rVert^{n-2}}$. What is meant with that? I do not know ...
0
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1answer
179 views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
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0answers
23 views

equivalent condition on being a polynomial

Let $f: \Bbb R^n \rightarrow \Bbb R$ be a continuous function. $X_f$ - subspace of $ \mathcal C( \Bbb R^n)$ generated by all translations of $f$, i.e. functions $ T_af(x) := f(x-a) $, $a \in \Bbb ...
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0answers
34 views

If $f(x)$ is concave, is $f(\lambda x) \le \lambda f(x)$ for $\lambda > 1$?

If $f(x)$ is concave, is $f(\lambda x) \le \lambda f(x)$ for $\lambda > 1$ ? A have a strong feeling that this should be the case. However, using the definition of concavity: $f(\alpha x + ...
8
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1answer
345 views

The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent

If a series $\sum\limits_{n=1}^\infty a_n$ is convergent, and $a_n\gt0$... Do not refer to Carleman's inequality or Hardy's inequality, show that the series $$\sum_{n=1}^\infty \frac ...
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5answers
321 views

How to prove that $\sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}}$ converges absolutely

If $\sum_{n=1}^{\infty}a_{n}$ converges absolutely, show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{1+a_{n}}$$ converges absolutely. My try: since $\sum_{n=1}^{\infty}a_{n}$ converges absolutely, then ...
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0answers
60 views

Prove that for a polynomial function $f$ if $f(x) \geq 0$ for all $x$, then $f(x) + f'(x) + \cdots + f^{(n)} (x) \geq 0$ [duplicate]

Prove that for a polynomial function $f$ with degree $n$ if $f(x) \geq 0$ for all $x$, then $f(x) + f'(x) + \cdots + f^{(n)} (x) \geq 0$. Give me some hints for this and please explain to me how you ...
2
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1answer
67 views

Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
0
votes
1answer
21 views

Best analytic upper bound for $(\log(x+1))^c$

I'm trying to get a good analytic upper bound for $(\log(x+1))^c$ in terms of $x$ for $x > 1$. An easy one comes from the fact that $\log(x+1) < 2\log x$, but $2^c (\log x)^c$ seems a bit ...
2
votes
1answer
42 views

Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
3
votes
1answer
97 views

Weierstrass product form

How to show the Weierstrass product form of the entire function $f(z)= \sinh z$ This question seem so interesting. I would like to write my some ideas, but I dont want to direct incorrectly. Please ...
3
votes
1answer
54 views

Heisenberg Bound

Question Verify $x(t) = e^{i\omega t}e^{-(t-\tau)^2}$ exactly satisfies the Heisenberg bound of $\sigma_t(x)\sigma_{\omega}(x)$. Attempt: I know $\sigma_t(x) = \int_{\mathcal{R}} ...
6
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1answer
151 views

A curious problem about Lebesgue measure.

The Problem: Let $(B(x_{m},0.5))_{m}$ be a sequence of disjoint open discs in $\mathbb{R}^{2}$ centered in $x_{m}$ and with radius 0.5. Let $\psi(n)$ be the number of these discs contained in the ...
2
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2answers
77 views

The conditions for Harmonics functions in complex analysis

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$. What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$. And I want to ...
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votes
1answer
39 views

Let $M$ be a subset of a Hilbert space $H,$ and let $v,w \in H$. Suppose that $\langle v,x\rangle=\langle w,x\rangle$ for all $x \in M$ implies $v=w$. [closed]

Let $M$ be a subset of a Hilbert space $H$, and let $v,w \in H$. Suppose that $ \langle v,x\rangle=\langle w,x\rangle$ for all $x \in M$ implies $v=w$. If this holds for all $v,w \in H$, show that $M$ ...
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2answers
100 views

How prove or disprove $f$ Non-differentiable points countably infinite

Today,my frend ask this follow question:and I consider sometime,and I can't solve it. I hope see someone can help me Question: let $f$ is continuous strictly increasing function, prove or ...
0
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1answer
66 views

Direct sum in $C[-1,1]$

(a) Show that the vector space $X$ of all real-valued continuous functions on $ [-1,1] $ is the direct sum of the set of all even continuous functions and the set of all odd continuous functions on ...
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2answers
31 views

Showing a limit for the mean value property

I need to understand a proof and i have a problem with the limit process at a specific point in this proof. I want to know, what is needed to get the following result: Let $f \in C^1(G)$, $\bar{x} ...
2
votes
1answer
62 views

Closure of numerical range contains spectrum

Let $A: D(A) \subset \mathcal{H} \to \mathcal{H}$ be a densely defined operator on a Hilbert space $\mathcal{H}$ with adjoint operator $A^{*}$. Given that $D(A) = D(A^{*})$ I'm trying to show that the ...
0
votes
2answers
74 views

A first-order non-linear ordinary differential equation containing $(f')^2$

The Problem: Find all differentiable functions $f: \text{D} \rightarrow \mathbb{R}$ satisfies: $$f(x)\left [ 1-f'(x)^2 \right ] = 2xf'(x) \; \forall x \in \text{D}$$ , whereas $\text{D}$ is a/an ...
2
votes
1answer
66 views

The multiplication formula for the Hurwitz/generalized Riemann zeta function

I'm having a difficult time showing that $$ \displaystyle \zeta(s,mz) = \frac{1}{m^{s}} \sum_{k=0}^{m-1} \zeta \left(s,z+\frac{k}{m} \right) $$ A couple of authors referred to it as an obvious fact. ...
0
votes
3answers
371 views

Compact sets of metric spaces are closed?

I am struggling with the idea that all compact subsets of a metric space are closed after reading chapter 2 of Rudin's Principles of Mathematical Analysis. The reason I am confused is that it seems ...
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0answers
56 views

Complex analysis visualization (Cauchy Theorem, Residue Theorem)?

I usually think of complex functions on the complex plane like vector fields. So basically what I have problems with is visualizing firstly Holomorphic functions. I have also read and successfully ...
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2answers
3k views

Pointwise vs. Uniform Convergence

This is a pretty basic question. I just don't understand the definition of uniform convergence. Here are my given definitions for pointwise and uniform convergence: Pointwise limit: Let $X$ be a ...
0
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1answer
57 views

Modulus of Continuity and Fourier Series

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ a $2\pi$-periodic function and define $\Omega(f,h)=|f(x+h)-f(x)|_{L^1}$. Show that exists a constant $C>0$ with $|\hat f(n)|\leq C.\Omega(f,h)$, where ...
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1answer
63 views

Show that if $f(1)=1$, then there exists a constant $\alpha$ such that $f(x)=x^\alpha$ for all $x \in (0, +\infty)$.

Let $f: (0, +\infty) \to\mathbb R$ be a differentiable function such that $f(xy)=f(x)f(y)$ for all $x,y \in (0, +\infty)$. Show that if $f(1)=1$, then there exists a constant $\alpha$ such that ...
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2answers
155 views

Uniform convergence of the series for $\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$

I am looking for the values where this series expansion converges uniformly. Intuitively, I believe the answer is $|x| < 1$, but I am not sure how I can show that using the Weierstrauss Majorant ...
0
votes
1answer
84 views

Find an upper bound on the absolute error of 3.141 as an approximation to π

Q: Find an upper bound on the absolute error of 3.141 as an approximation to π I have no idea what to do... :( What I know: absolute error = real value - approximate value Help :)
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1answer
72 views

Proving a continuous function $f:K \cup A \to \mathbb R$ is uniformly continuous if $K$ is compact and $A$ is discrete.

Let $(X,d)$ be a metric space. Let $K \subset X$ compact and $A \subset X$: $\exists \delta>0$ such that $d(a,b)>\delta$ for all $a,b \in A$ with $a \neq b$. Consider in $K \cup A$ the induced ...
5
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2answers
338 views

Prove that $f(x) = x^{1/5}$ is continuous everywhere

Need to prove that $f(x) = x^{1/5}$ is continuous everywhere, where $f: \mathbb{R} \to \mathbb{R}$: from definition we need to show that given $ \epsilon > 0 $ $\exists \delta > 0 $ s.t. ...
1
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1answer
96 views

How prove $f=0$,if $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}=f$

Question: let $D$ is Bounded closed region,and Assume $f$ is Continuous differentiable on $D$,if such $$\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial y}+\dfrac{\partial f}{\partial ...
3
votes
1answer
99 views

Why is the derivative the tangent vector?

I'm trying to understand, at least intuitively why the derivative of a function at a point is the tangent vector at this point. If we see the functions of this form $f:\mathbb R\to \mathbb R$ we see ...
0
votes
1answer
35 views

Measureability of Simple Functions over different sets.

Suppose E and F are subsets of R (we do not know yet whether E and F are Lebesgue measurable). Suppose we also know the function $s(x) = 5\chi_{E} (x) + 2\chi_{F} (x) is Lebesgue measurable. Prove ...
3
votes
1answer
57 views

Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
0
votes
2answers
37 views

Prove $f(x) \rightarrow l \ $ as $ \ x \rightarrow a \ \iff \ f(x_{n}) \rightarrow l \ $ as $ \ n \rightarrow \infty \ $, ($a$ is a limit point)

Suppose $ S \subset \mathbb{R}^{N} \ $ and $f: S \rightarrow \mathbb{R}^{M} \ $ Let $a$ be a limit point of $S$ and let $l \in \mathbb{R}^{M}$. Prove $f(x) \rightarrow l \ $ as $ \ x \rightarrow a \ ...
4
votes
2answers
67 views

Solve the integral? [closed]

Help me? please How to solve this integral? $$\int\frac{1+x^2}{1+x^4}\,dx$$