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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2 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
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2answers
18 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
20 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
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 ...
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1answer
20 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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1answer
48 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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0answers
21 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 ...
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2answers
43 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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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 ...
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2answers
42 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?
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0answers
30 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
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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 ...
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0answers
15 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
41 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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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 ...
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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 ...
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0answers
13 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
23 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
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
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1answer
26 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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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 ...
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1answer
38 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
14 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
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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?
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0answers
14 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
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5answers
332 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?
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2answers
35 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 ...
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1answer
21 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
37 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' = ...
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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
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3answers
79 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 ...
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0answers
10 views

Looking for modern equivalent of Whittaker and Watson.

I am looking for a modern treatment of transcendental functions with an emphasis on difficult calculations similar to the classic text by Whittaker and Watson (now over 100 years old) ...
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1answer
29 views

Verification of a delta/epsilon Proof of continuity

So I am asked to show that $f:\mathbb R\rightarrow \mathbb R$ is strictly increasing and $f^{-1}: f(\mathbb R)\rightarrow \mathbb R$ is continuous at $1$. My $f(x)$ is a point wise function $$f(x) = ...
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1answer
39 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
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1answer
23 views

True or false statement about a $C^\infty$ function

If $f \in C^\infty$ and $f^k(0)=0 \forall$ $k \in \mathbb N \bigcup \{0\}$,then can we conclude that $f$ is identically zero?
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2answers
26 views

A Question on Bolzano-Weierstrass Theorem

The following are two equivalent statements about the Bolzano-Weierstrass theorem. Every bounded real sequence has a convergent subsequence. A subset of $\mathbb R$ is compact if and only if it is ...
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1answer
25 views

Limit of a continuously differentiable function that statisfies

Let $x(t)$ be a continuously differentiable for all $t>0$, and such that: $$\lim_{t\to \infty}[x'(t)+x(t)]=\alpha$$ I need to show that $\lim_{t\to \infty}x(t)=\alpha$ My goal is to show that ...
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1answer
26 views

Proving that the solution to the IVP exists given a condition on the right hand side

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$. Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$. In order ...
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21 views

Limit of Solution of an ODE

Consider the following differential equation: $y^{'}(t)=g(y)$ where $g$ is a continuous function from $\mathbb{R^n}$ to $\mathbb{R^n}$. Assume that $y(t)$ is a solution to the previous ODE. Suppose ...
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2answers
17 views

Convergence of functions in a metric space

Let $C([0,1])$ be the space of all continuous functions from $[0,1]$ to $\mathbb{R}$ under the metric $$ \lVert f \rVert_1 = \int_0^1 \lvert f(x) \rvert \, dx. $$ Now consider $f_n(x) = e^{-nx}$. I ...
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0answers
12 views

$\|\nabla f\|_p\leq C (\|\nabla \times f \|_p +\|\nabla \cdot f\|_p)$

Let $f\colon\mathbb{R^3}\to \mathbb{R^3}$ have compact support. The identity $$ -\Delta = \nabla\times\nabla \times - \nabla \nabla \cdot, $$ and two integration by parts shows that $$ \|\nabla f\|_2 ...
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1answer
37 views

Torsion vs Curvature of a curve

I was wondering if there is a simple explanation of the torsion and curvature in $\mathbb{R}^3$ of a curve. The curvature measure somehow the acceleration perpendicular to the tangent vector, but ...
2
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1answer
43 views

Inverting a function

I am stuck with the following problem I am supposed to find the inverse of the function $g$ with $2$ variables, where $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
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2answers
85 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
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1answer
16 views

Closed Interval in $E^2$

I am currently working through Introduction to Analysis by Rosenlicht In one of the exercises $4.30,$ he asks a question regarding a closed interval in $E^2.$ I am not sure what this means. I was ...
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1answer
11 views

Measure associated to locally integrable function is regular

Suppose $f \in L^{1}_\text{loc}(\mathbb{R}^{d})$ with $f \geq 0$ and create the measure $\mu$ such that $d\mu = f\, dx$ where $dx$ is the standard Lebesgue measure. Must $f$ be a regular measure? ...
0
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0answers
12 views

Assumptions on the Borel measure in Stein's Harmonic Analysis

I am currently reading the proof of Theorem 1 on Page 13 of Stein's Harmonic Analysis which proves that if $f \in L^{1}(\mathbb{R}^{n})$, then for every $\alpha > 0$, $$\mu(\{x: (Mf)(x) > ...
4
votes
2answers
90 views

Find the limit $\,\,\, \lim_{n \to \infty}\Big(\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\Big)^{1/n} $

What is the limit of: $$ \lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big) \cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}? $$ By computer, I guess the limit is equal to ...
0
votes
1answer
23 views

$\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$\{0}

I need to prove the set identity of the complex logarithm $\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$. ...