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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1answer
9 views

Integration without complex analysis on rational-improper integral

Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Without the use of complex-analysis. With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis?
1
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1answer
34 views

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum's ...
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0answers
17 views

The condition of uniform convergence of $\sum a_n\sin(nx)$

If $a_n$ satisfy: $a_n \geq a_{n+1}$, and $a_n \rightarrow 0$ as $n \rightarrow +\infty$, show that: $$\sum_{n=1}^{\infty}a_n\sin(nx)$$ is uniform convergence in $\Bbb{R}$ if and only if $$\lim_{n ...
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0answers
16 views

Convergence of distances in metric space [on hold]

If $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are Cauchy sequences in $(X,d)$. How do i show that $(a_n):=d(x_n,y_n)$ converges?
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1answer
27 views

The cardinal of the set of all measures on $\mathbb{R}$

It is a very simple question that I don't know how to do: Let $M = \{\mu \colon \mathcal{B}(\mathbb{R})\to \mathbb{R} \colon \mu \text{ is a measure}\}$ $$|M| = \ ?$$ Any help will be appreciated.
1
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1answer
34 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
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0answers
24 views

Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
0
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1answer
29 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
2
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2answers
30 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
2
votes
3answers
151 views

a limit property at infinity

Let $k\in(0,1)$ is fixed and $L$ is a finite value. Is it possible to say if $\lim_{x\to\infty}f(x)=L$ then $\lim_{x\to\infty}f(kx)=L.$
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1answer
43 views

Proving least upper bound property implies greatest lower bound property

In Rudin 1.11 Theorem Proof he claims the following Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of ...
2
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3answers
106 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
2
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1answer
30 views

Predicting the increase/decrease of number

I have these entries in my database that looks like this: ...
2
votes
1answer
39 views

Associated Legendre polynomials

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
3
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5answers
92 views

Complex Analysis book including integration

FOR BEGINNERS: Currently, I am looking for a textbook on complex analysis, which covers complex analysis from the beginning, and majorly focuses on contour integration, and the residue theorem. On ...
1
vote
1answer
27 views

A function differentiable only at $0$ and for $|z|=1$

I need to find a polynomial function that is differentiable at the origin where $f'(0)=1$ and at every point $|z|=1$ but at no other point in the complex plane. I just have no clue how to solve ...
3
votes
3answers
91 views

Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
2
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0answers
46 views

Properties of the Fourier transform

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. I want to show that $$ \widehat{g}(kn)= \widehat{h}(k), \\ \widehat{g}(l)=0, l \not\equiv 0 \ \text{mod} \ n.$$ I ...
0
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0answers
37 views

Complex analysis (Analytic function, sharp upper bound)

I encouter complex analysis problems the I think it is quite to do. Could anyone please give a hint or guideline. Thank you very much in advanced. Let $D$ be an open unit disc $\{z \in \mathbb{C}| ...
2
votes
1answer
47 views

$f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki

Paraphrased from Munkres' Topology: Lemma 9.2. Given a collection $\mathcal{A}$ of nonempty sets, there exists a choice function \begin{equation*} f: \mathcal{A} \to \bigcup\limits_{A \in ...
2
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5answers
86 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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0answers
54 views

How to prove that an injection from a sphere into a Euclidean space is homotopic to a constant?

How to prove that the injection $i: S^{m-1}\rightarrow \mathbb{R}^m$ is homotopic to a constant ? Where $S^{m-1}=\{x\in \mathbb{R}^m, |x|=1\}$ Thank you.
2
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1answer
25 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
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vote
4answers
52 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
1
vote
1answer
77 views

Does $\int_{0}^{1}x^nf(x)\, dx=0$ imply that $f=0$ a.e. without assuming $f \in C[0,1]$?

Suppose that $f \in L^{1}[0,1]$ and $\int_{0}^{1}x^nf(x)\, dx=0$ for $n=0,1,2,\dots$ Does that imply that $f=0$ a.e.? I think that there will be a counterexample but it is hard to find out.
0
votes
1answer
25 views

differentiabilty implies continuity (analysis)

Is my proof correct? We need to show that if $f$ is differentiable at $x_o$, then it is continuous at $x_o$ i. e. $$\forall \epsilon >0, \exists \delta >0 \text{ s.t. } ...
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0answers
41 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
0
votes
1answer
29 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
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1answer
48 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
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0answers
29 views

Variational Inequalities and how they are used?

I am doing undergrad research in this field next semester and I have never heard of this topic before. I tried wikipedia and reddit for help but nothing seems to help. I just want to know what I'm up ...
1
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1answer
24 views

Limit of arc-length of a curve

Let $L(f)$ denote the length of a curve $f$, if $f = \lim\limits_{n\to\infty} f_n$ then do we necessarily have that $L(f) = \lim\limits_{n\to\infty} L(f_n)$? I assume that we will have some continuity ...
4
votes
3answers
266 views

Image of open set is not open?

I'm confused by the proof that $\epsilon$-$\delta$ continuity is equivalent to open-set continuity. One can prove that a function is $\epsilon$-$\delta$-continuous if and only if the preimage of any ...
0
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1answer
41 views

Hint on metric space [on hold]

I want show that $(a_n):=d(x_n,y_n)$ converges, if $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are cauchy sequences in $(X,d)$. Here is what i do; From the hypothesis, $(a_n)$ is bounded, because ...
6
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3answers
83 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
0
votes
1answer
41 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
3
votes
1answer
42 views

Radius of convergence, prove that $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely for every $z \in \mathbb{C}$ with $|z| < R$

A proof of this is given in my lecture notes as follows: We define $R$ to be $\sup \{|z| \in \mathbb{R} : \sum |c_k z^k|$ converges $\}$ when the supremum exists. Prove that $\sum |c_k ...
3
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1answer
46 views

periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$

Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$ I want to discusse about non-constant periodic solution of it. Can someone give a hint that how to start to think. And does it have ...
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0answers
25 views

statistical analysis [on hold]

Two independent random samples of annual starting salaries for individuals with masters and bachelors degrees in business were taken and the results are shown below ...
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2answers
21 views

Proving Lipshitz continuous over a convex set with Projection Operator

Suppose a problem $$\min_{x \in \mathbb{R}^{n}} f(x)$$ subject to $x \in \Omega$ which is a closed and convex set. If $\nabla f(x)$ is Lipschitz continuous in $\Omega$, then prove that $$e(x) = x - ...
0
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1answer
34 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
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2answers
57 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
1
vote
1answer
97 views

Given $|f'(x)|\leq r<1$ show that $f(x)=x$ is unique solution

Suppose that $|f'(x)|\leq r<1, \forall x\in R$. How do I show that the equation $f(x)=x$ has a unique solution?
2
votes
1answer
34 views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write ...
4
votes
1answer
42 views

Differential equation: $A(x)y''(x)+A'(x)y'(x)+y(x)/A(x)=0$

So give the differential equation $$A(x)y''(x)+A'(x)y'(x)+\frac{y(x)}{A(x)}=0,$$ with $A(x)$ a known function and $y(x)$ te be determined. What is the solution for this differential equation ? I've ...
2
votes
3answers
106 views

How to compute the integral $\int^{\pi/2}_0\ln(1+\tan\theta)d\theta$? [duplicate]

How to compute the integral $\int^{\pi/2}_0\ln(1+\tan\theta)d\theta$. If we let $t=\tan\theta$, then the integral becomes to ...
1
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0answers
18 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
0
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0answers
26 views

Inhomogeneous ODE (2nd order) - question to Laplace-transformation?

I've the following inhomogeneous second order ODE: $$a_1\cdot u(t) + a_2\cdot u'(t) + a_3\cdot u''(t) = b_1\cdot y(t) + b_2\cdot y'(t) + b_3\cdot y''(t)$$ The parameters $a_i$ and $b_i$ are ...
1
vote
3answers
62 views

Showing instablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
2
votes
0answers
23 views

Weak convergence in Lp [on hold]

got a little problem with this ex. I could use some help. Let $U := \Pi_{i=1}^d(a_i, b_i) \subset \mathbb{R}$ ($a_i < b_i$ for each $i$) and let $f \in L^p(U)$ for some $1<p<+\infty$. Let us ...
2
votes
1answer
44 views

Is the polynomial a zero polynomial?

Let $p(x)$ be a polynomial over $\mathbb{R}$ with $deg[p(x)]\leqslant n$. If $p(1)=p(2)=\cdots = p(n+1)=0$, then will the polynomial be necessarily a zero polynomial? i.e., if a polynomial of degree ...