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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51 views

Alternative proof of de l'Hospital theorem

I see that the usual proof of de l'Hospital theorem involves the mean value theorem and is carried on in a case-by-case fashion. Could you point out some other proofs of de l'Hospital that are ...
0
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1answer
16 views

Help with complex analysis series question that's troubling me

-What does it mean for a series to be 'absolutely' convergent? -what does the notation (Zn)n denote? Struggling with this question any help is appreciated bu a solution would be much appreicated! ...
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29 views

How to (dis)prove the following about harmonic functions

I've been asked the following and really don't know where to start... Prove or disprove the following: If $u$, $v \colon \mathbb{R^{2}} \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate ...
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2answers
48 views

Surjection of the map $f:\mathbb R^{2} \to \mathbb R^{2}$

Let $f:\mathbb R^{2} \to \mathbb R^{2}$ be given by $f(x,y)=(x+y,xy)$.Then, which are correct? $f$ is surjrctive. The inverse image of each point in $\mathbb R^{2}$ under $f$ has atmost two ...
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3answers
61 views

Some questions in series of functions uniform convergence

I am just solving many question on series of uniform convergence in order to get more intuition and experience with this stuff so I wanted to know what do you guys think of my argument. Its easy for ...
2
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0answers
39 views

Refinement of Lebesgue Decomposition Theorem

On Wikipedia, a "refinement" of the Lebesgue decomposition theorem is given, and it is also given as problems in Stein and Shakarchi and Bruckner and Thomson. Can someone provide a comprehensive proof ...
0
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1answer
24 views

Uniform convergence of series argument

Hey guys so I have the following series. $\sum x^2/(1 + n^2x^2)$ and asked of it will uniformly converge or not on R. I am mean I see it that it will diverge but I can't see a formal argument that I ...
2
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0answers
39 views

Particular $L^p$ space

I am confusing some definitions. Suppose we have a Cauchy sequence $(f_n) \subset L^2(\Omega,C^0([0,1],\mathbb{R}))$, where $\Omega$ is a measurable space with measure $\mu$ and ...
3
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1answer
29 views

Prove $\{g(x_n)\}_{n=1}^\infty$ converges

Let $g : (a, b) → R$ be uniformly continuous on $(a, b)$. Let $\{x_n\}_{n=1}^\infty$ be a sequence in $(a, b)$ converging to $a$. Prove that $\{g(x_n)\}_{n=1}^\infty$ converges. The general idea here ...
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0answers
24 views

tensor products of Banach space

Let $E_{1},\cdots, E_{n}$ be Banach spaces; $n\in\mathbb{N}$ and $\mathbb{R}$ be a real numbers and $E\widehat{\otimes}\mathbb{R}$ be a completion tensor product. We have the fact that ...
2
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0answers
70 views

MVT and functions

Let $f$ be defined on an open interval $I := (a,b)$. (a) Let $x$ and $y$ be real numbers such that $x<y$. Show that if $z \in [x,y]$, then there is some $t \in [0,1]$ such that $z=tx+(1-t)y$. (b) ...
2
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1answer
37 views

Integral form of this IVP

How do I show that the following initial value problem $$ xu''+u'+xu=0,\quad u(0)=1,\quad u'(0)=0 $$ has the following integral form: $$ u(x)=1+\int_{0}^{x} t\ln(t/x)u(t)\,dt $$ I am stuck because if ...
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2answers
17 views

Prove that an alternating series converges to a value between the first term and the sum of the first two terms

This is used in an elementary proof that $e$ is irrational. I can prove this, but what I am doing is not particular nice looking. In the proof, the author says this is obvious but I can't seem to ...
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0answers
22 views

Some intuition for this question of uniform convergence

Let $f_n(x) = nx / e^{nx}$ for $x \in [0,3].$ b)Show that convergence is not uniform on $[0,3]$ c)Let 0 < t < 3 . Determine on which interval [0,t] or [t,3], the convergence is uniform. Okay ...
0
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1answer
56 views

Convergence of $a_n = (1+i)^n+(1-i)^n$

I am asking a question related to Is there a formula for $(1+i)^n+(1-i)^n$? I am looking on the exact same term, just as a sequence, so i want to find out: Is $a_n = (1+i)^n+(1-i)^n$ convergent or ...
0
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1answer
29 views

Proof of harmonic conjugates being symmetric

Prove or disprove the following: If $u, v :\mathbb R ^2 \to\mathbb R$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, show whether or ...
2
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3answers
64 views

How to show that $\tan(n), n\in \mathbb{N}$ is not bounded

I'm struggling on how to show that the sequence of $\tan(n)$ is not bounded. Can you please give me some help?
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0answers
22 views

Format argument for uniform convergence

Hi guys so I have this question about uniform convergence that I solved but I wanted to get to see of how rigorous is my argument and there could be done any improvements. $Let f_n(x) = nx / (1 + nx) ...
4
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1answer
56 views

Determine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$

Detemine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$. My attempt: Upon attempting to use the Weierstrauss M-test I get ...
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1answer
45 views

Show that it is Lipschitz

Let $E \subset \mathbb{R}^d$ be Lebesgue measurable und let $\phi (t)=m \left ( \Pi_{i=1}^{d} (-\infty , t_i ) \cap E \right )$. I have to show that $\phi $ is Lipschitz. Could you give me some ...
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0answers
34 views

onto and strictly increasing implies homeomorphism

I have this question, let $f:[\alpha,\beta[\rightarrow [a,b[$ an onto map and strictly increasing , how to prove that $f$ is an homeomorphism, it means $f$ and $f^{-1}$ are continuous ? Thank you
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39 views

Basic proof, but just need confirmation that its ok!

I am reading Tao's Analysis 1. I am asked to prove that n x m = 0 iff n=0 or m=0. Please keep in mind that multiplicative inverse has not been defined in my text so i don't want to use it! If noone ...
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1answer
92 views

How to prove this integral-inequality.

Suppose $f$ is twice differentiable and satisfies $f(0)=0$. Prove the inequality. $$\int_0^1 |f(x)f'(x)| dx \le\ \frac{1}{2} \int_0^1 |f'(x)|^2 dx $$ This is a problem from undergraduate math ...
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1answer
23 views

Show g' is increasing and deduce the inequality

Let $g :=$ -ln $x$ where $x>0$. Show $g′$ is increasing on $(0,∞)$ and deduce that if $n ∈ \mathbb{N}$, $x_1,...,x_n$ are positive and $s_1, . . . , s_n$ are non-negatives with $\sum_{k=1}^n s_k = ...
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0answers
32 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
4
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1answer
40 views

Sturm-Liouville problem and periodic boundary conditions

I was wondering about this: I know that if a 1-d Sturm-Liouville operator is limit circle or limit point then the eigenvalues are simple ( so no degenerated spectrum). But in the case of periodic ...
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1answer
30 views

Does it stand that $\lim \inf |f_n|^p=|f|^p$ for this reason?

We have that $f_n, f \in L^p, 1 \leq p < +\infty$, $f_n \rightarrow f $ almost everywhere and $||f_n||_p \rightarrow ||f||_p$ . Do we have that $\lim \inf |f_n|^p=|f|^p$ because of the following?? ...
0
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1answer
20 views

Closed function and adherence

I have by definition that a function is cloded if the image of a closed set by this function still closed And i want to prove that $$f:E\rightarrow F ~\text{is closed} \Leftrightarrow ...
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1answer
23 views

Characterizing conditions for $\tanh{(kx-b)}=x$ to have 1/2/3 fixed points.

I am trying to understand what are the conditions for $\tanh{(kx-b)}$ to have 1 or 2 or 3 fixed points. That is I am trying to characterize conditions on $k$ and $b$ for which equation ...
2
votes
1answer
18 views

How can the derivative of the Euclidean norm be exhibited without considering partial derivatives?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be given by $f(x) = \|x\|$. I would like to show that $Df(a)h = \frac{a\cdot h}{\|a\|}$ without resorting to using partial derivatives. I considered the ...
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1answer
33 views

Does $(a+b)^p \leq 2^{p-1}(a^p+b^p)$ stand also when we have minus?

It stands that $$(a+b)^p \leq 2^{p-1}(a^p+b^p)$$ Does it also stand when we have $"-"$ instead of $"+"$ ?? Does it stand that $$|f_n-f|^p \leq (|f_n|^p+|f|^p)$$ ?? ($f_n, f \in L^p, 1 \leq p ...
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2answers
26 views

Convergence of sequences in $\mathbb{C}$

I am currently getting into the field of complex numbers, with the imaginary unit $i^2 = -1$ and stuff. At the moment i am looking onto a few sequences in $\mathbb{C}$, regarding convergence. I have ...
0
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1answer
44 views

Function as derivative of some other function

I know we can't write every function as derivative o some other function. Darboux theorem gives easy way to find such example. And I just read absolutely continuous function which gives condition when ...
4
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2answers
72 views

Prove that $f $ is constant

Let $f:\mathbb R \to \mathbb R $ be a continuous function such that for all $x \in \mathbb R$, $f(x)=f(x^2) $ prove that $f$ is constant. "please give me hints not answer. thanks a lot. :):):):):)" ...
2
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1answer
60 views

given $\epsilon >0$ , there is an $A\in \mathfrak{a}$ with $\overline \mu (A-E)+\overline \mu (E-A)<\epsilon$

Let $\mu$ be a measure on an algebra $\mathfrak{a}$ and $\overline \mu $ the extension of it given by the Caratheodory process. Let $E$ be measurable with respect to $\overline \mu $ and $\overline ...
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3answers
718 views

Integration substitution: How does Wolfram Alpha come up with this step?

I have to integrate $$ \int \frac{1}{(\sin x) (\cos x)} \, dx $$ I looked at the Wolfram Alpha step by step solution to figure out how to do it. First, it rewrites the integral as: $$ \int (\csc ...
0
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1answer
27 views

Determine whether or not the convergence is uniform

Let $f_n(x) = x(1 - x)^n$. Find $f(x) = lim \: f_n(x)$ Determine or not the convergence is uniform on [0,1]. Okay so first of all we find that the point wise limit is zero and this function we expect ...
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0answers
14 views

Definition of Measure regular

Book's, Real and complex analysis, Walter Rudin. I am somewhat confused. My question is: "In other words, we are looking at $L^p$, where $\mu$ is Lebesgue measure on $[0,2\pi]$(or on $T$), ...
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1answer
52 views

Is this derivative somehow bounded?

I have a function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is a test function and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by $f(x) = \phi(\frac{\|x\|}{n})$. Now if I take any ...
2
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1answer
40 views

Approximate $C^{\infty}$ functions by test functions in the Sobolev space norm

I am looking for a way to approximate a function $f \in \mathbb{C}^{\infty} \cap H^m(\mathbb{R}^n)$ by test functions such that I approximate $f$ and all of $f's$ $m-$ derivatives in the canonical ...
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2answers
59 views

Investigate whether there exists a function $f : [a,b]\rightarrow R$ that is continuous and that takes exactly twice each of its values. [duplicate]

Let a < b. Investigate whether there exists a function $f : [a,b]\rightarrow R$ that is continuous and that takes exactly twice each of its values.
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18 views

Show that for any $\epsilon> o$ there exists a *step function* $s:[a; b] \rightarrow R$ such that |f(x)-s(x)| < for all $x\in I$.

Let $a < b$ and $f : [a; b] \rightarrow R$ be a continuous function. Show that for any $\epsilon > 0$ there exists a step function $s:[a; b] \rightarrow R$ such that $|f(x)-s(x)| < ...
0
votes
1answer
22 views

find a counterexample to show it is not bounded

Let $a < b$ and $f : (a,b) \rightarrow R$ be a continuous function that is locally bounded. Does it follow that f is bounded on (a,b)? Give arguments for your answer. I was thinking that it ...
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2answers
38 views

Bounded and discontinuous proof

Problem: Suppose that $f : [a, b] \to \mathbb{R}$ is bounded and discontinuous at exactly one point $c$ between $a$ and $b$. Prove that $f$ is $\mathbb{R}$-integrable. I know that $f$ is bounded and ...
2
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0answers
27 views

Initial conditions to solve an ODE?

Given is the following inhomogenous linear ODE (4th order): $$q_0\cdot\sigma + q_1\cdot \dot\sigma + q_2\cdot \ddot\sigma + q_3\cdot \dddot\sigma + q_4\cdot\ddddot\sigma = p_0\cdot\epsilon + ...
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3answers
88 views

Assume $f(x)$ has continuous derivative on $[0,1]$ ,$|f'(x)| \leq M$

Assume $f(x)$ has continuous derivative on $[0,1]$ with $|f'(x)| \leq M$ and $f(0)=f(1)=0$. Prove that $$\left| \int_0^1 f(x) \ dx \right| \leq \frac{M}{4}$$
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3answers
58 views

What is essence of second order derivative?

Let $u(x)\in C^2(R)$ is a real function. so: $$ u'(x)=\lim_{\Delta x\rightarrow 0} \frac{u(x+\Delta x)-u(x)}{\Delta x} $$ And: $$ u''(x)=\lim_{\Delta x\rightarrow 0} \frac{u'(x+\Delta x)-u'(x)}{\Delta ...
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0answers
27 views

Mobius Function and Liouvlle's Function

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...
0
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1answer
24 views

Finding Maximum of Symbolic Function of Fourth Derivative Using MATLAB

I am trying to find the maximum $y$-value of the fourth derivative of the function $f(x) = \frac{1}{1.1+cos(x)}$. I am limited to real numbers. I know that the answer should be 6100 and occurs at ...
2
votes
1answer
73 views

Baire's property iff first category has dense complement.

Show that $(S, d)$ has Baire's property iff every set of first category has a dense complement. A set is of first category if it is a countable union of nowhere dense sets. First Category Baire's ...