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
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1answer
14 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
0
votes
0answers
20 views

having trouble with take $\displaystyle\lim_{\alpha \to -1}$

$\Gamma(0)=\infty$ I have : $$\displaystyle \lim_{\alpha \to ...
-1
votes
0answers
13 views

In the geometrical interpretation for integration how lower and upper rectangular approximation are functions of natural number?

I've attempted to prove this in the following manner. Let Q be a subset of $P[a,b]$ which contains partitions of each order exactly once. Now, if we consider mappings $F:N \to Q$ defined by $F(n)=p$ ...
-2
votes
1answer
22 views

Reposting Question about Schroder-Bernstein

Assume there exists a $1$-$1$ function $f:X\to Y$ and another $1$-$1$ function $g:Y\to X$. Follow the steps to show that there exists a $1$-$1$, onto function $h:X\to Y$ and hence $X\sim Y$. a) The ...
0
votes
1answer
15 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
2
votes
0answers
35 views

Solve complex integral with $\Gamma$-function [on hold]

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
0
votes
0answers
22 views

On the continuity of Li's numbers.

Consider Li's numbers defined by $\lambda_n = \sum_{\rho} \left(1-\left(1-\dfrac{1}{\rho}\right)^n\right)$ where $n$ is a nonnegative integer and the $\rho$ are the nontrivial zeros of the Riemann ...
0
votes
2answers
38 views

Decide what is the number of roots of the equation

Decide what is the number of roots of the equation $2^x=100x$. I know I can draw a sketch and then check but maybe there is a better method to do that? It's an exam question, thus it must require ...
1
vote
2answers
39 views

Convergence of $\int_2^{\infty}f(x)\,dx$ with a given condition

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
0
votes
1answer
13 views

Find the distance such that the angle will be the gratest

Rectangle shaped screen in a cinema is 8m high. It is place on a wall in such a manner that the upper edge of the screen is 12m above the floor. Find the distance between the viewer and the wall where ...
0
votes
2answers
69 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
0
votes
0answers
13 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
0
votes
0answers
24 views

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...
0
votes
3answers
66 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$ [on hold]

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
-1
votes
0answers
20 views

Find minimum distance between the plane and the beginning of Cartesian plane.

Find minimum distance between the plane: $S=\{\left(x,y,z\right) \in \mathbb{R}^3: x+yz=2012 \}$ and the beginning of Cartesian plane $(0,0,0)$. I want to minimize this with use of lagrange's ...
1
vote
0answers
28 views

Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to x_0} f(t)=f(x_0)$

The problem: Suppose $f:[a,b]\to R$ and define $g: [a,b]\to R$ as follows: $g(x)=\sup \{f(t):a\le t\le x \}$ Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to ...
5
votes
1answer
79 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
0
votes
2answers
30 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
-1
votes
0answers
16 views

Is the following derivative also differentiable with respect to $n$?

Let $f(x)$ be the $n-th$ derivative with respect to $x$ of $x \exp (n-1) log (x-1)$ evaluated at $x=1$. Is $f(x)$ differentiable with respect to $n$ ?
4
votes
3answers
88 views

Is the $n$th derivative of a continuous function also continuous?

Consider a differentiable (and hence continuous) function of order $n-1$. Is the $n$th derivative of such a function always continuous? As an example, is the $n-th$ derivative of the function $f(s) = ...
0
votes
1answer
20 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
1
vote
1answer
21 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
5
votes
2answers
76 views

Can $\mathbb R$ be partitioned into a countable number of dense subsets with same cardinality?

Is it possible to partition $\mathbb R$ into an countable number of disjoint dense subsets with the same cardinality? Furthermore, is it possible to partition the reals into an uncountable ...
2
votes
1answer
29 views

Every ordered field that has the least upper bound property is isomorphic to the real number system.

Okay, so here's a theorem from Rudin: "Every ordered field that has the least upper bound property is isomorphic to the real number system." Here's a definition: "Ordered fields are isomorphic if ...
0
votes
0answers
19 views

Big $O$ question for While and For loops [on hold]

I have to find the exact $O(N)$ for these instructions, not just the order of magnitude. I'm not getting any of the answers provided for me. I know the first loop is $O(3N+2)$. The declaration of ...
2
votes
1answer
38 views

Show that $ \int_0^2 e^{x^2-x} dx \in [2e^{-1/4},2e^2] $

Show that $ \int_0^2 e^{x^2-x} dx \in [2e^{-1/4},2e^2] $ If $f(x)\leq g(x)$ for $x\in[a,b]$ then $\int^b_af(x)dx\leq \int^b_ag(x)dx$ if $x\in [0,2]$ then $x^2-x\leq x$, so $$0 \leq \int_0^2 ...
1
vote
6answers
59 views

Limit of $\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$ [on hold]

Limit of $$\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$$ I know (thanks to wolfram) it is equal to $y^2$, but i do not know how to show that.
-1
votes
2answers
30 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
1
vote
2answers
23 views

Finding a minimum of a function, measuring the sum of the squares of distance from some points of the $\mathbb{R}^n$

Given are a finite number of points $a_1, ..., a_m \in \mathbb{R}^n$. Consider the sum of the squares of distance: $$f(x) = \sum_{k=1}^m ||x-a_k||^2, x \in \mathbb{R}^n$$ with $||.||$ being the ...
0
votes
2answers
41 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
0
votes
0answers
12 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
1
vote
2answers
50 views

Calculate double integral $\iint_A \sin (x+y) dxdy$

Calculate double integral $$\iint_A \sin (x+y) dxdy$$ where: $$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$$ How to calculate that? $x+y$ in sin is confusing as i do not ...
3
votes
2answers
45 views

Double integral $\int\int_A y dx dy$

Calculate Double integral $$\iint_A y dxdy$$ where: $$A=\{(x,y)\in\mathbb{R}^2 : x^2+y^2\le4, y \ge 0 \}$$ I do not know what would be the limit of integration if i change this to polar coordinates. ...
-1
votes
1answer
26 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
-1
votes
2answers
30 views

A Crucial Observation On Li's Criterion for the Riemann Hypothesis?

In 1997, Xian Jin Li formulated an interesting criterion whose validity is completely equivalent to the Riemann Hypothesis, namely: Define the real number $(n-1)!\lambda_n$ to be the $n-th$ ...
0
votes
1answer
22 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
0
votes
0answers
5 views

Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
0
votes
1answer
14 views

Classify the growth of functions and find a more general growth function

The following function $f(t,x):[0,T]\times R\mapsto R$ such that $\int^T_0|f(t,0)|^2 d t<\infty$, where $0<T<\infty$. If $f(t,x)$ satisfies $|f(t,x)|\leq Ax+B$ for each $x\in R$ and $A, B$ ...
1
vote
1answer
61 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
0
votes
0answers
23 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
3
votes
0answers
40 views

Ramanujan Infinity sum functional equations

i was reading about the mellin transform ans i found the following $$\sum _{k=1}^{\infty } \left(\frac{e^{-k x}}{e^{-2 k x}+1}-\frac{\pi \text{sech}\left(\frac{\pi ^2 k}{x}\right)}{2 ...
1
vote
0answers
29 views

What powers of $|x|$ belong to $L^1$?

Prove that $|x|^ {−qp} \in L^{1}(U)$, where $U=B_{1}(0)\subset \mathbb{R}^{n}$. I think I could use polar coordinates to facilitate the work but not sure if it is useful.
1
vote
1answer
18 views

uniform continuity

Let $F(s,y)$ be uniformly continuous in $[a,b] \times B$, where $B \subset R^n$ is a closed subset. Assume $x_k \rightarrow x$ in $C[a,b]$ with $x_k(t) \in B$ and prove $$\int_a^b F(s,x_k(s)) ds ...
7
votes
2answers
104 views

Real analytic functions

I'm writing because I don't know the usefulness of real analytic functions. I mean, I know that analyticity is something more respect differentiable ($C^\infty$ function), but I don't have in mind a ...
0
votes
1answer
22 views

Exercise: Uniform Boundedness Principle and Double dual

Let $X$ be a normed vector space and $(x_{n})$ be a sequence in $X$. Show that if the sequence $f(x_{n})$ is bounded for every $f \in X^{\ast}$, then there exists $C > 0$ such that $\|x_{n}\| < ...
1
vote
0answers
24 views

Check pointwise convergence and uniform convergence of $f_n(x) = n^3x\exp(-nx^2)$ on $[0,1]$

Check pointwise convergence and uniform convergence of $f_n(x) = n^3x\exp(-nx^2)$ on $[0,1]$ Pointwise convergence: $$ \lim_{n\rightarrow\infty} f_n(x) = \lim_{n\rightarrow\infty} ...
4
votes
3answers
55 views

When can a set have an upper bound but no least upper bound?

So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'. I don't understand how a ...
2
votes
2answers
34 views

Injectivity of the function $x||x||$ on $\mathbb R^n$

Let , $f:\mathbb R^n\to \mathbb R^n$ be a function defined by $f(x)=x||x||^2$ for $x\in \mathbb R^n$. Then , which are correct ? (A) $f$ is one-one. (B) $f$ has an inverse. Here $f$ is not a ...
1
vote
0answers
21 views

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continous on $(0,\pi)$

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continuous on $(0,\pi)$ I think about showing the uniform convergence of $$ f_k: \mathbb (0, \pi) ...
0
votes
0answers
9 views

Reference of integral on differential manifolds and conformal aplications

I need goods and fast reference about integral of differential manifolds, more precisely about results of change variable but not with differential forms. I need goods and fast reference about ...