3
votes
2answers
52 views

Finding $f(x)$.

If $$f(x)=1+x+x^2+\displaystyle\int_{0}^{x}e^k f(x-k) dk$$ then how do we find the function $f(x)$? Is there a way to solve it, with or without arriving at a differential equation? This a homework ...
0
votes
0answers
27 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
1
vote
1answer
27 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
1
vote
0answers
27 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
0
votes
0answers
20 views

Describing an open interval I centered at c, $I \subseteq (a, b)$

Entire question: Let (a,b) be an open interval of Real numbers and let $c \in (a,b).$ Describe an open interval I centered at c such that $I \subseteq (a,b)$ I didn't quite get where I should've ...
1
vote
0answers
32 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
0
votes
0answers
31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
-1
votes
0answers
18 views

Cardinality of the following set of functions on $\mathbb R$ [duplicate]

Consider the following set $W$ = The set of constant functins on $\mathbb R$. $X$ = The set of polynomial functins on $\mathbb R$. $Y$ = The set of continous functins on $\mathbb R$. $Z$ = The set ...
0
votes
0answers
46 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
1
vote
2answers
47 views

Find out the interval where Rolle's Theorem is applicable

Find out the interval for which the Rolle's theorem is valid for the function $f(x)=2x^3+x^2-4x+2$ My attempt : Supposing the interval is $[a,b]$, $f(a)=f(b)$ gives the equation ...
0
votes
1answer
23 views

Small question about limit at $+\infty$ to $-\infty$

please the definiton of $\displaystyle\lim_{u\rightarrow+\infty}G(t,u)=-\infty$ is: $\forall M>0 ,\exists R>0 $ such that $|u|\geq R \Rightarrow G(t,u)\leq M$ or $G(t,u)\leq -M$ ? please ...
2
votes
0answers
63 views

Problem in functional analysis.

I heard of this problem that caught my attention and I am curious now thus I would appreciate if I could have a hint or a solution. Let $(x_n)$ a sequence in a normed space $X$ such that ...
6
votes
3answers
242 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
5
votes
3answers
108 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
6
votes
5answers
180 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
0
votes
1answer
42 views

Questionable Intervals

Find the numbers $X_1 , X_2 , \ldots , X_{10} $ such: $X_1$ is in the interval $[0,1]$. If we divide the interval $[0,1]$ in halves,each half consists of only one of $X_1$ or $X_2$. If we divide ...
1
vote
3answers
36 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
2
votes
1answer
45 views

Are These Two Definitions of a Disconnected Set Equivalent?

I found two definitions of a disconnected set $E \subset \mathbb R$. $E$ is disconnected if: (1) there are disjoint open sets $A, B$ such that $A \cap E$, $B \cap E \ne \emptyset$, and $(A \cap E) ...
1
vote
2answers
44 views

$E \subset \mathbb R$ is an Interval $\iff E$ Is connected

My text gives the definition that $E$ is disconnected if there exist disjoint open sets $A, B$ such that: $A \cap E$, $B \cap E$ are nonempty. $(A \cap E) \cup (B \cap E) = E$. Then for the ...
4
votes
1answer
75 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
0
votes
1answer
49 views

is bounded partial derivative continous

Let $f:{\mathbb R}^2\rightarrow {\mathbb R}$ be defined as: $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2}, & \ (x,y)\ne(0,0),\\ 0, & \ (x,y)=(0,0).\\ \end{array} \right. $$ Prove ...
4
votes
4answers
135 views

Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
4
votes
1answer
58 views

Is it true that $\lim_{x\to a}f(x)=0$ if and only if $\lim_{x\to a}|f(x)|=0$?

Is it true that $\lim_{x\to a}f(x)=0$ if and only if $\lim_{x\to a}|f(x)|=0$? I intuitively think this is true, but really no idea to prove it. Can you give me hints?
1
vote
0answers
25 views

A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu $ and $\lambda $ are measures in $M$ st $\mu \ge $ $\lambda $ then show that $m$ defined as $\mu= \lambda + m $ is a measure 2) Prove that if ...
2
votes
2answers
36 views

Proove of equality of integrals

I'm currently sitting on the following problem: Let f be in the set of the integrable functions(:=$L^¹(\mathbb{R}^n))$, A $\in \mathbb{R}^{n\times n}$ invertible. Therefore define g:=$\mathbb{R}^n ...
9
votes
2answers
108 views

Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
0
votes
0answers
38 views

Find the Equation of the Tangent Line (Real-Analysis)

Let $f: \mathbb R \rightarrow \mathbb R$ be defined by $f(s) = {1 \over \pi} \int^s_0 e^{-t^2} dt$. Find the equation of the line tangent to the graph of $y = f^{-1}(x)$ at $x=0$. I left out the ...
0
votes
3answers
81 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
1
vote
2answers
76 views

Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
0
votes
2answers
40 views

Study of a function

I have this function $\displaystyle g(s)=\frac{s^{2-\sigma}}{1+s^2}, ~\text{for all} ~s\in \mathbb{R}$ , i need to find the interval of $\sigma$ and the maximum of the function $g$. I calculate the ...
0
votes
2answers
48 views

Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases} $$ doesn't admit weak solutions. I'm proceeding by ...
1
vote
1answer
34 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
3
votes
1answer
116 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
3
votes
1answer
111 views

Derivative of the parameter

I have the equation$\begin{cases} x'(t)=x(t)+y(t) \\y'(t)= \mu y^2(t)+x(t)\end{cases}$ Cauchy problem $\begin{cases} x(0)= 1 + \mu \\y(0)=-2\end{cases}$ . I must calculate $\frac{\partial ...
1
vote
0answers
39 views

problem of computing limit

The problem is to prove the following for $n \geq 3$ $$u(0)=\frac{1}{n\alpha (n) r^{n-1}}\int_{\partial B(0,r)} g dS +\frac{1}{n(n-2)\alpha (n)} \int_{B(0,r)} (\frac{1}{|x|^{n-2}} - ...
1
vote
1answer
27 views

Pointwise and uniform convergence of sequence of functions

Let $(f_n)$ be a sequence of continuous functions on $\mathbb R$. If $(f_n)$ converges to $f$ pointwise on $\mathbb R$, then $$\lim\limits_{n\to ...
0
votes
0answers
25 views

Find Jordan decomposition constructed by function.

$$ F(x) =\left\{ \begin{array}{l l} 0 && x \leq 0\\ 1-x && 0 \leq x \leq 1\\ 2x-4 && 1 < x \leq 2 \\ 1 && 2 < x \end{array} \right. $$ ...
1
vote
2answers
35 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
3
votes
2answers
54 views

Convergence radius of power series is infinite

Which function is given by a power series whose convergence radius is infinite? $$A. \ \ \ e^{-\frac{1}{x^2}}$$ $$B. \ \ \ \sin{\left(\frac{1}{x}\right)}$$ $$C. \ \ \ ...
0
votes
0answers
20 views

Identity proof vector calculus

I encounter massive problems tackling the following math problem. Especially the notation in line #2 with the $\frac{\partial u}{\partial x^2}$ and in line #3 the $ U:]0,R[\times\mathbb R\ $ , ...
9
votes
1answer
119 views

Convergence of $\sum_{n=1}^{\infty}\frac{1}{3^n\ \sin(n)}$

Does this series converge? Root test and ratio test are inconclusive.
1
vote
3answers
58 views

Let $f:(0,1) \rightarrow \mathbb R$ be continous. Suppose that $|f(x) - f(y)| \leq |\sin x - \sin y |$ for all $x,y\in (0,1)$. Then

$f$ is discontinuous at least one point in $(0,1)$ $f$ is continuous everywhere on $(0,1)$, but not uniformly continuous on $(0,1)$ $f$ is uniformly continuous on $(0,1)$ $\lim_{x \rightarrow 0^+} ...
0
votes
3answers
52 views

Show that series converges

Show that if $ \{ p_n \} $ is a Cauchy sequence, then it has a subsequence $ \{ p_{n_k}\} $ such that the series $ \sum_{k=1}^\infty b_k $ converges, where $ b_k = d(p_{n_k}, \, p_{n_{k+1}}) $. My ...
0
votes
1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
1
vote
1answer
49 views

Show that series in Cauchy Sequence

Let $a_n = d(p_n, p_n+1)$ for $n = 1, 2,\cdots $. Show that if the series $\displaystyle \sum^{∞}_{n=1} a_n$ converges, then $\{p_n\}$ is a Cauchy sequence. My Approach: I thought of using the ...
0
votes
1answer
18 views

Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is M path connected?

The Assignment: Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is S path connected? Explain your answer. I don't think S is path-connected since ...
1
vote
1answer
48 views

proof of coarea formula for n dimensional hypersurface in $R^n$

$f:R^n \rightarrow R$ be continuous and summable. please give the proof for these formulas $\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$ $\frac{d}{dr}\int_{ ...
1
vote
4answers
77 views

Proving that a sequence is monotone and bounded

Let $x_1> 1$ and let $x_{n+1} := 2 - \displaystyle\frac{1}{x{_n}}$ for $n \in \mathbb{ N}$. Show that $(x_n)$ is bounded and monotone. Find the limit. I am confused on how to show that the ...
0
votes
3answers
67 views

Find $f(t)$ such that $ f\left(\dfrac{t^2}{2-t}\right) = -5t+4 $.

Find $f(t)$ such that $ f\left(\dfrac{t^2}{2-t}\right) = -5t+4 $. I don't really know how to approach this problem. Would you give any hint on how to start?