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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24 views

Prove that f(x) is regulated.

Define $f:[0,1]\to \mathbb{R}$, $f(x):=0$ if $x\notin \mathbb{Q}$, $f(p/q):=1/q$, $q>0$, $p, q$ coprime integers. Prove that $f$ is regulated. A function $f:[a,b]\to\Bbb R$ is a regulated ...
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1answer
12 views

solving equations with log and polynomials.

I need to solve/estimate for x in the following equation - $Klnx + x^\beta = r$. $K,r > 0.$ An estimate for large r(fixed K) is what I am looking for.
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1answer
22 views

How prove this Ratio Test and Its Generalizations problem?

Question: let $\alpha\in (0,1)$,and the postive sequence $\{a_{n}\}$ such $$\lim_{n\to\infty}\inf \left(n^{\alpha}\left(\dfrac{a_{n}}{a_{n+1}}-1\right)\right) =\lambda\in (0,+\infty)$$ show ...
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0answers
16 views

How is the interchange of the limit and the maximum valid at this point in Erwin Kreyszig?

In 1.5-5 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, the author shows completeness of the space $C[a,b]$ of all (real- or complex-valued) functions defined and continuous ...
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0answers
23 views

Give an example of an open bounded set whose boundary has no (Lebesgue, Jordan) measure zero

Calculus (believeitornot!) homework due Tuesday. They already give me a hint: consider an open cover of intervals in Q∩[0,1] whose "lengths sum up shortly!".
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1answer
13 views

Another Riesz-ian Question Regarding Chapter 2 of Rudin's Real and Complex

So as to save space/time, the proof is detailed here: http://data.imf.au.dk/kurser/advanalyse/F06/lecture16-print.pdf My question is why is it that the measure of compact $K$ (in page 17 of source, ...
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0answers
20 views

boundary of a 3-cell

Let $I^k=[0,1]^k$. I want to calculate $\partial (I^3)$ rigorously. In case of $I^2$, one can easily separate as $\partial I^2 =\partial\sigma_1+\partial\sigma_2$, where $\sigma_1=[0,e_1,e_2]$ and ...
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3answers
39 views

sum of infinite series with telescopes

I need help finding the sum of the infinite series $$\sum_{k=1}^\infty \frac{1}{n(n+1)(n+2)}$$ I have used the partial fraction decomposition to get this as the sum of ...
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1answer
16 views

Real 2D Analysis Question using Brouwer's Fixed Point Theorem

My question is as above. Currently I am stuck at the very start, part (i)! I can't come up with an appropriate $f$, even though I've been thinking about it for ages! If someone would be able to ...
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0answers
29 views

Taylor Series Expansion for Function of Two Variables (with Countable Discontinuities)

Given a real-valued function of two real variables, under certain conditions of smoothness in a closed ball about some point, we can obtain a Taylor series for the function about that point. I want ...
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2answers
45 views

What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
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1answer
16 views

Prove concavity without testing the second derivative..

Consider a function $F(L)=(L-L^2a)T^{L-1}$, where $0<L<\frac{1}{a}$. The constants $a$ and $T$ may take values over $]0,1[$ and $[0.01,0.1]$, respectively. The first derivative of $F$: ...
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1answer
17 views

Show that if a subset of an ordered field has a least upper bound, then any two least upper bounds for it have to coincide

Should I prove by contradiction then turn my proof into a direct proof using contrapositives? That seems unnecessarily complicated..
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1answer
25 views

Existence and uniqueness for the ODE $y''-y^{1/3}=0$

For the ode $ y'' - y^{1/3}=0 $, is there any way to check the existence and uniqueness of the solution? I know the Picard's Theorem, but it can only be used for the first order ode.
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2answers
35 views

Prove that there is an integer $N$ such that $\frac{N}{10^k} \leq x \lt \frac{N+1}{10^k}$ [on hold]

I am having difficulty with this problem. Any help will be greatly appreciated. x is an element of the real numbers. k is an element of the natural numbers. Problem: Prove that there is an integer ...
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0answers
31 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
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2answers
33 views

what kind of b should be if $\cos(bx)=\sin^2x+1$

If $$\cos(bx)=\sin^2x+1$$ has no solutions other than $x=0$, then b should be: rational or irrational??? Tried: let $b=1$ then the equation holds whenever $x=2k\pi$. So b should be irrational...
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1answer
32 views

$\{p_{n}\}$ is a sequence of real numbers. Prove $\limsup$ $\{p_{n}\} < \infty$ if and only if $\{p_{n}\}$ is bounded above.

I have done the following. $\Leftarrow$ $\limsup$ $\{p_{n}\}$ is the set of suprema of all the subsequential limit points of $\{p_{n}\}$. So, if it were not finite, then, given any $M\in N$, ...
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0answers
19 views

Name for hypergeometric-like sum.

Consider the sum $F(a_1,\cdots,a_r; z)\sum_{m=0}^{\infty}\frac{(a_1+m)(a_2+m)\cdots(a_r+m)}{(a_1)(a_2)\cdots (a_r)}\frac{z^m}{m!},$ where $a_i$ for $i=1,2,\cdots,r$ is an increasing sequence of ...
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2answers
51 views

Prove by mathematical induction that exponentials grow faster than polynomials

How to prove that for $\forall q>1$ $\forall k\in \mathbb{N}$ $\exists c>0$ $\forall \in \mathbb{N}$ $q^n≥cn^k$? I should use mathematical induction.
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1answer
37 views

How did Fourier series lead to the development of rigorous analysis?

Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him ...
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0answers
12 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
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1answer
19 views

Unifrom Convergence and the weierstrass M test

$\sum_{n=1}^\infty\frac{1}{{1+n^{2}x^{2}}}$ on the interval x $\in$ [1,$\infty$] I tied to apply weierstrass m test but i didn't find any Series that R = $\infty$ how can i apply weierstass m test ...
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1answer
18 views

Composition of a function with a metric

Which of the following functions, $f: [0,\infty) \rightarrow [0,\infty)$, can be composed with a metric $d$ to get a new metric $f \circ d$: a)$\;f(x) = \begin{cases}0 & \text{if $x=0$} \\x+1 ...
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1answer
17 views

Limit of sequence involving a product

This question is related to a post that was deleted. I want to calculate the following limit $$\lim_{n \to ...
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1answer
19 views

How do I get an Archimedean spiral that decreases from an initial radius?

So, where the equation of an archimedean spiral is: $$r = a + b\theta$$ I want to be able to use the equation in this form to then have a function where r decreases in exactly the same way by an ...
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0answers
18 views

Approximating an L2 function by analytic functions

Spose I have a $ h \in L^2(U) $ where $ U \subset \mathbb{R}^3 $ is open and bounded. Is it possible to approximate this by analytic functions? If so, spose now we take $ U = \mathbb{R}^3 $. Is this ...
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1answer
18 views

How to prove that f(x)=1/|x-t| is continuous but not bounded?

Suppose S is not closed: there is a point t in R, t not in S, such that a sequence in S converges to t. Show that the function f: S-> R, defined by f(x) = 1/|x -t|, is continuous but not bounded.
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1answer
28 views

Cantor's intersection Theorem without the diameter hypothesis

In proving Cantor's in intersection theorem, the fact that limit of the diameter of the sets is 0 was used to prove that the intersection is non-empty. I just wondered if that hypothesis is excluded ...
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2answers
30 views

By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
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2answers
28 views

Show that $(g\circ f)':[a,b] \to R$ is differentiable and find the derivative. (Chain Rule Proof)

I know I probably shouldn't ask two questions in a short amount of time but this is a rather simple one. Question: Suppose $f : [a, b] \to [c, d]$ and $g : [c, d] \to \mathbb{R}$ are differentiable ...
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1answer
12 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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5answers
433 views

Prove that g is differentiable

Question: Suppose f, g, and h are defined on (a,b) and $a < x_0 < b$. Assume f and h are differentiable at $x_0$, $f(x_0) = h(x_0)$, and $f(x) \le g(x) \le h(x)$ for all x in (a,b). Prove that g ...
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Proof of taking derivatives of both equal sides

I am curious about the proof of the following or whether the statement is true in general Assume that I have the following property: $f(x,y)=g(x,y,z)$ Can I assert that $D_xf=D_xg$ at any point ...
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1answer
47 views

Uniform convergence

Determine whether or not the given se ries of functions converges uniformly on the indicated interval (set) $\sum_{n=1}^\infty\frac{(1)}{(nx)^2}$ where x $\in (0,1]$ I don't know if we can apply can ...
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1answer
14 views

invertibility, derivative, and difference quotient

Suppose that $f$ is an invertible differentiable function, that the domain of $f^{-1}$ contains an interval around $a$, and that $f^{-1}$ is continuous at $a$ and that $f^{-1}$ is continuous at $a$. ...
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1answer
36 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
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1answer
25 views

A basic ergodic question

I know that irrational number can be approximated by p/q and error less than 1/q^2. But I still cannot give a rigorous proof to this problem. And how to show that the difference between the left ...
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3answers
26 views

how to find the smallest s to make f continuous at (0,0)

$$ f(x,y)=\left\{ \begin{array}{lll} \frac{|x|^s|y|^{2s}}{x^2+y^2} & \text{if}& (x,y) \neq (0,0)\\ 0 & \text{otherwise} \end{array} \right. $$ what is the smallest s to make f(x,y) ...
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0answers
17 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
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1answer
22 views

Find $f$ such that the contraction $\phi$ has a fixed-point $\rho= \sqrt{2}$

I use the Newton method and the Banach fixed-point theorem and have: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous and $f: I \rightarrow \mathbb{R}$ a ...
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2answers
15 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
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1answer
53 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{n→\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
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0answers
17 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
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1answer
40 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.
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11 views

composition of Riemann integrable functions.

I have two functions: f:[a,b]->R and g:[c,d]->R where a My question is if it follows that g o f (the composite function of f,g) is Riemann integrable as well?
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2answers
77 views

Find this maximum of this $\frac{\int_{0}^{\pi}f(x) \, dx}{\int_{0}^{\pi} f(x)\sin x\,dx}$

Question: Assmue that $\int_0^\pi f(x)\,dx$ and $\int_0^\pi f(x)\sin x\,dx$ is convergence,and $f(x)>0,\forall x\in(0,\pi)$ Find this maximum as possible for all function $f$ ...
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0answers
21 views

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$?

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$ ?. I know that zero is an essential singularity, and so $f(D(0,\delta)-\{ 0 \})$ is dense in $\mathbb{C}$. This question ...
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0answers
50 views

What is known about$\sum\limits_{p\text{ prime}} \frac{1}{p^2-1}$?

Are there some known results for $\sum\limits_{p\text{ prime}} \dfrac{1}{p^2-1}$? I wasn't able to find anything about this sum in the internet or in my books!
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0answers
30 views

A complicated question

I have the following operator $A: H^1_{0,p}\longrightarrow H^1_{0,p}$ be defined by \begin{equation} Au(t)=\int_0^{+\infty} G(t,s)q(s)f(s,u(s))\,ds-\sum_{k=0}^{+\infty}G(t,t_k)h(t_k)I(u(t_k)), ...