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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3answers
24 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) ...
0
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0answers
16 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
0
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1answer
19 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 ...
0
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2answers
14 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 ...
1
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1answer
41 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 $\sum a_k$ ...
0
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0answers
15 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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0answers
9 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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1answer
29 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 $f$ ...
2
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0answers
41 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
29 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)), ...
2
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0answers
16 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
0
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1answer
18 views

$\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$

I can't compute this $\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$. I have separate it into 2 integrals but i can't continue.
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1answer
16 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
1
vote
1answer
31 views

How find this sum $\sum_{n=1}^{\infty}\frac{n}{[(2n)!]^2}$

Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{n}{[(2n)!]^2}$$ I think we can note $$\dfrac{n}{((2n)!)^2}=\dfrac{1}{2}\dfrac{2n}{((2n)!)^2)}=\dfrac{1}{2}\cdot\dfrac{1}{(2n)!\cdot(2n-1)!}$$
1
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0answers
23 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
3
votes
1answer
21 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
2
votes
1answer
29 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
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0answers
6 views

A branch of the argument in the domain $\mathbb{C}$\ {$te^{it}|t\geq0$}

Usually most questions I come across on branches have domains similar to $\mathbb{C}$\ $[0,\infty)$, $\mathbb{C}$\ {$te^{i\alpha}|t\geq0$}. But the domain $\mathbb{C}$\ {$te^{it}|t\geq0$} is ...
2
votes
2answers
28 views

Calculate an integral that has a sum within.

Im trying to calculate this integral: $\displaystyle \int_{0}^{\pi} \sum_{n=1}^{\infty} \frac{n \sin(nx)}{2^n}$ The only thing I have been able to do is switch the integral and the sum, and in the ...
0
votes
2answers
22 views

Example of $x$ being adherent point but not accumulation point?

So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an ...
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0answers
19 views

What can the Weierstrass therem say about an arbitrary continuous function being analytic?

Please describe the conditions and why they do not match up, as clearly not every continuous function is analytic. Thank you!
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2answers
46 views

The product of uniformly continuous functions is not necessarily uniformly continuous

I was asked to show that given two functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ which are both uniformly continuous, to show that the product ...
0
votes
2answers
30 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
1
vote
1answer
23 views

Fejer's theorem with Riemann integrable function

If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then $$ \lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = ...
1
vote
2answers
65 views

What branch of analysis deals most with sequences and series?

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what ...
0
votes
0answers
24 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
3
votes
2answers
49 views

Tedious undefined limit without L'Hospital $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$

When I try to calculate this limit: $$\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$$ I find this: $$\begin{array}{l} L = \mathop {\lim }\limits_{x \to ...
0
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0answers
16 views

Showing a fact about $\sigma$-algebras and Borel sets

Let $(\Omega,\mathcal{A})$ be a measurable space, $(A_n)_{n\in\mathbb{N}}\subset\mathcal{A}$ and $f_n:\Omega\to [-\infty,\infty]$ be a $\mathcal{A}-\overline{\mathcal{B}}$ measurable function, where ...
1
vote
2answers
46 views

$\int_1^e \! \frac{\mathrm{d}x}{x\,\sqrt{-\ln \ln x}}$

$$\int\limits_1^e \! \frac{\mathrm{d}x}{x\,\sqrt{-\ln \ln x}}$$ I can't find any antiderivative, is it possible to calculate the definite integral?
-1
votes
0answers
35 views

MERGE(L1,L2) two sorted list [on hold]

Merge(L1, L2), below, is a function which merges two sorted linked lists L1 and L2, and outputs a single sorted linked list. By "sorted", we mean increasing order. ...
2
votes
0answers
21 views

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k ...
0
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0answers
21 views

Uniform convergence on subintervals

(a) Fix a positive integer $M$ and let $\{f_{n} : [0, M]\rightarrow \mathbb{R}\}$ be a sequence of functions. Suppose that $f_{n}\rightarrow f$ pointwise on $[0, M]$ and that $f_{n}\rightarrow f$ ...
0
votes
2answers
42 views

How to find the limit without L'Hospital rule

Find the limit $$\lim_{x\to\infty} x\left[\left(1+\frac{1}{2x}+\frac{1}{4x^2}+\frac{1}{8x^3}\right)^{1/3}-1\right].$$ I assume that L'Hospital rule here is useless and something else must be done.
-1
votes
0answers
13 views

Analysis, about Riemann-intrability [on hold]

Consider the following function $$ f(x) = \begin{cases} x & \text{if } x \in \mathbb{Q} \cap [0, 1] \\ x^3 & \text{if } x \in \mathbb{Q}^c \cap [0, 1] \end{cases} $$ Is the ...
0
votes
1answer
21 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
0
votes
0answers
16 views

Riemann Integrable on Circle implies $L^2$ Convergence of linear interpolation.

I was reading the proof of well-known theorem. If $f$ is Riemann integrable on $[0,2\pi]$, then its Fourier series converges to $f$ in $L^2$ norm. The proof was depending on the $L^2$convergence ...
-1
votes
1answer
25 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
0
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2answers
33 views

How prove $0\le \lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{i+\frac{1}{i}}-\ln{\frac{n}{\sqrt{2}}}$

let $$x_{n}=\dfrac{1}{1+1}+\dfrac{1}{2+\frac{1}{2}}+\cdots+\dfrac{1}{n+\frac{1}{n}} -\ln{\dfrac{n}{\sqrt{2}}},n=1,2,\cdots$$ show that $a=\displaystyle\lim_{n\to\infty}x_{n}$ is exsit,and $0\le ...
1
vote
1answer
29 views

Fundamental solution of heat equation on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold of $m$ dimensional. Then there exists a sequence $(\phi_i, \lambda_i)_{i\in\mathbb{N}}\subset C^\infty(M)\times\mathbb{R}_{\geq0}$ such that ...
0
votes
0answers
18 views

If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...
0
votes
1answer
39 views

Discontinuity of the identity function in topology

According to a theorem I was taught, the identity function $id(x)=x$ from $(\mathbb{R}, \tau_1)$ to $(\mathbb{R}, \tau_2)$ is continuous if $\tau_1 = \tau_2$. Are there any examples of topologies ...
0
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0answers
15 views

proving properties of Lagrange-function

Let $L:\mathbb R^n\times\mathbb R^n\to\mathbb R$ be $$L(q,x):=\Psi(x)f(|q|)-\Phi(x)$$ where $f\in C^2([0,\infty))$ such that $f'(0)=0$ and $f''\geq\delta>0$ on $[0,\infty)$ for some constant ...
0
votes
2answers
29 views

Limit by L'hospital's rule

I have to prove that: $$\lim \limits_{x \to \infty} \frac{{\int_x^{\infty} \exp(-t^2/2)dt}}{\exp(-x^2/2) (1/x)}=1$$ Should I use L'hospital rule, if yes what are the derivatives?
1
vote
0answers
19 views

Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. ...
2
votes
5answers
336 views

How calculate limit without using L'hospital theorem

$$\lim_{x \to 0}\left(\frac{\sin{x}-\ln({\text{e}^{x}}\cos{x})}{x\sin{x}}\right)$$ My question is: This limit can be calculated without using the L'Hospital theorem? any ideas?
0
votes
2answers
36 views

A strange 3rd order ODE

This is the original ODE: $ y^{1/2}y'''+e^{-x}(y'')^{2+c}-(\frac{xy}{x+1})y'=x $ with c is a positive number. $y(0)=1,y'(0)=0,y''(0)=1$ $1st$ question: If x is large, then $ y^{1/2}y'''$ and ...
3
votes
1answer
22 views

checking slope = $0$ at a point for a function using $\epsilon $, $\delta $ definition

From the continuity definition, a function is continuous at a point $a$ if : $$\forall \epsilon \gt 0 \exists \delta \gt 0 : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$$ If I change the ...
2
votes
0answers
39 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
1
vote
0answers
24 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
2
votes
3answers
82 views

If $f'(x)=f(x)+\int_{0}^{1}f(x)\,dx$ and $f(0) = 1,\,$ then what is the value of $\, \int_0^1 f(x)\,dx=$?

If $\displaystyle f'(x)=f(x)+\int_{0}^{1}f(x)\,dx\,$ and $\,f(0) = 1.$ Then what is value of $\displaystyle \int f(x)\,dx\,?$ $\bf{My\; Try.}$ Let $\displaystyle \int_{0}^{1}f(x)\,dx = A\;,$ Then ...