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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59 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
616 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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1answer
62 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
19 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
38 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
37 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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2answers
38 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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1answer
29 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
18 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
82 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
46 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
124 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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2answers
110 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
57 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
56 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
45 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)), ...
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0answers
41 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 ...
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1answer
29 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
28 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 ...
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1answer
36 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 $$\dfrac{n}{((2n)!)^2}=\dfrac{1}{2}\dfrac{2n}{((2n)!)^2)}=\dfrac{1}{2}\cdot\dfrac{1}{(2n)!\cdot(2n-1)!}$$
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1answer
70 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
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1answer
79 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 ...
3
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1answer
155 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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2answers
33 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 ...
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2answers
87 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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2answers
187 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 ...
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2answers
73 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 ...
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1answer
40 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} } = ...
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2answers
152 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 ...
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2answers
80 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 ...
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2answers
52 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?
2
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3answers
56 views

Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$

Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$. Is the first step here to notice that $$ \frac{1}{z-2} + \frac{1}{z-3} = ...
2
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1answer
36 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 ...
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0answers
42 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 $f_{n}\rightarrow f$ ...
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2answers
59 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.
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1answer
49 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 ...
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0answers
32 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 ...
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1answer
53 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 ...
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2answers
37 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 ...
2
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1answer
79 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 ...
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1answer
54 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 ...
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0answers
19 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 ...
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3answers
48 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?
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0answers
86 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. ...
3
votes
5answers
464 views

How can I calculate this limit without using L'Hopital's rule?

$$\lim_{x \to 0}\left(\frac{\sin{x}-\ln({\text{e}^{x}}\cos{x})}{x\sin{x}}\right)$$ Can this limit be calculated without using L'Hopital's rule?
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2answers
52 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
30 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
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0answers
76 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' = ...
4
votes
2answers
110 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
89 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 ...