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).

learn more… | top users | synonyms (1)

1
vote
1answer
102 views

Local Maximum of the integral over an unknown function

this problem is a part of my project but currently I don't have a very good idea to solve it. Here is the function: $Z(t) = k\mathrm{e}^{-bt} \int_0^t I(\tau)\mathrm{e}^{b\tau}\mathrm{d}\tau$ where ...
1
vote
1answer
119 views

analysis double series sums

Let $a_{ij}$ be entries in a matrix in $i$th row and $j$th column such that $a_{ij} = \left\{ \begin{array}{l l} 0 & \quad \text{if $i$ < $j$}\\ -1 & \quad \text{if $i$ = $j$}\\ ...
4
votes
2answers
513 views

Convergence/Divergence of Sequence defined by a recurrence relation

Given the following sequence: $$ a_{n+1} = a_n(2 - a_n) $$ for which values $a_1 \in \mathbb{R}$ does this sequence converges or diverges. By trial and error I found that for $a_1 \in (0, 2)$ it ...
7
votes
1answer
100 views

How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$?

Assume $f:[a,b]\to[a,b]$ be continuous and differentiable on $(a,b)$ and $f(a)=a$, $f(b)=b$. How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$? Thanks in advance.
1
vote
0answers
56 views

Complex Analysis - confusion over conjugate

Suppose we denote $\overline z = z^*$. Then in polar form, $z^*$ = $re^{-i\theta}$. Now, we know that $e^{z}$ is an entire function $(z=x +iy)$ So: 1) Why is $\large \displaystyle \frac{z}{z^*} = ...
1
vote
1answer
113 views

Write down an equation for paths with the following image sets

i) The triangle with the corners $1,-1+i,-1-i$ where the initial point and final point is $1$. ii) The union of two circles with radius $1$ and centers $i$ and $-i$, where the initial and final point ...
5
votes
1answer
816 views

Is $\mathbf{R}^\omega$ in the uniform topology connected?

Let $\mathbf{R}^\omega$ be the set of all (infinite) sequences of real numbers. Then is this space connected in the uniform topology? How to determine this? The uniform metric $p \colon ...
1
vote
3answers
102 views

Evaluating a complex integral over a half-ring

I need to integrate the $z/\bar z$ (where $\bar z$ is the conjugate of $z$) counterclockwise in the upper half ($y>0$) of a donut-shaped ring. The outer circle is $|z|=4$ and the inner circle is ...
0
votes
2answers
58 views

How to determine if this map is open or closed?

Given two supspaces $X:= [0,1]\cup[2,3]$ and $Y:=[0,2]$ of $\mathbf{R}$, let $f \colon X \to Y$ be defined as follows: $$f(x):= \left\{ \begin{array} {ll} x & \mbox{if $0\leq x\leq 1;$} \\ x-1 ...
2
votes
1answer
261 views

Justification behind changing coordinates of a differential operator

On many websites focused on physics, (say http://skisickness.com/2009/11/20/ ) they like to represent differential operators in different coordinates. I.e. going from the standard basis to polar ...
1
vote
1answer
100 views

About Convergence of the Image of a Convergent Sequence Under a Uniformaly Convergent Sequence of Functions

Let $X$ be a topological space and $Y$ a metric space. Let $f_n \colon X \to Y$ be a sequence of continuous functions. Let $x_n$ be a sequence of points of $X$ converging to a point $x \in X$. Suppose ...
6
votes
0answers
137 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
2
votes
1answer
4k views

Orthogonal polynomials and Gram Schmidt

How can we use the Gram Schmidt procedure to calculate $L_0,L_1, L_2, L_3$, where ${L_0(x), L_1(x), L_2(x), L_3(x)}$ is an orthogonal set of polynomials on $(0, \infty)$ w.r.t. the weight function ...
0
votes
1answer
64 views

Write down the equation for the paths with the following image sets

i)The straight line connecting the points $1+i$ and $3-i$ where $1+i$ is the initial point and $3-i$ is the final point. ii)The circular arc with the initial point $3+i$, final point $1+3i$ and ...
1
vote
1answer
159 views

how compute $\int_a^b xf(x)dx$? such that$ f(a+b-x)=f(x)$

let $f(a+b-x)=f(x)$ then how compute $$\int_a^b xf(x)\,dx$$ thanks for any hints
4
votes
1answer
221 views

Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book)

So I was looking at the proof given in Bott, Tu "Differential Forms in Algebraic Topology" of how to approximate continuous mapping by smooth mappings between manifolds. It is Proposition 17.8 on Page ...
15
votes
3answers
297 views

To what extent is the taylor polynomial the best polynomial approximation?

Given a function $f\in\mathscr C^n([a,b])$ and a point $x_0\in [a,b]$, to what extent is the n-th taylor polynomial $T_n(x,x_0)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$ the best polynomial ...
0
votes
2answers
352 views

Product of open set is open?

Show that $A\in \mathbb R^{n+m}$ is open IFF for each $(x,y)\in A$, with $x\in \mathbb R^n, y\in\mathbb R^m$, there exist open sets $U\in\mathbb R^n, V\in \mathbb R^m$ with $x\in U, y\in V$ such that ...
2
votes
0answers
160 views

Step functions dense in Integrable functions with respect to $L_2$

Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
2
votes
4answers
85 views

How to prove $\{y\in\mathbb R^n: |x-y|=r \text{ for some }x\in X\}$ is closed for closed $X$ and fixed positive $r$?

Let $X\subset\mathbb R^n$ be a closed set and $r$ a fixed positive real number. Let $Y=\{y\in\mathbb R^n: |x-y|=r \text{ for some }x\in X\}$. Show that $Y$ is closed. I tried to approach this problem ...
2
votes
1answer
209 views

Uniform convergence in the Poisson equation…

Let $K$ a compact set and $(u^j)_j$ a sequence of functions bounded uniformly such that $$\Delta u^j=f^j(u^j), \ \ \ \mbox{in} \ \ K,$$ where the sequence $(f^j)$ is bounded uniformly and ...
0
votes
1answer
638 views

Using Leibniz Integral Rule on infinite region

I am trying to take the derivative with respect to $a$ of some function $I(a)=\int_{0}^{\infty}f(a,x)dx$. I would like to make sure that I am using the Leiniz Integral Rule correctly. Various web ...
3
votes
0answers
319 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
0
votes
1answer
200 views

Monotone increasing sequence?

For which values of $x\in\Bbb R$ is the sequence $a_n:=\left(1+\frac xn\right)^n$ monotone increasing? Thanks in advance!
2
votes
1answer
67 views

For which continuous functions does this hold?

For which continuous functions $f:(a,b)\to\Bbb R$ is it true that $\forall x,y\in\Bbb R:f(x+y)\le f(x)+f(y)$ ? Thanks
2
votes
3answers
310 views

Norm of bounded operator on a complex Hilbert space.

It is fairly easy to show that for a bounded linear operator $T$ on a Hilbert space $H$ $$||T||=\sup_{||x||=1,||y||=1}|\langle y, Tx \rangle |.$$ If $H$ is a complex Hilbert space, can you show that ...
1
vote
1answer
39 views

Question about integrable functions

Is it true that if $f: X \rightarrow \mathbb R$ is integrable on a measure space $(X, M, \mu)$ then for arbitrary $\varepsilon >0$ there exists a set $A \in M$ with finite measure such that ...
1
vote
2answers
740 views

Natural logarithm limit

Is $$\lim_{n\rightarrow +\infty}\ln\left(\frac{n+1}{n}\right)=0?$$ Because it is $\ln(1+\frac{1}{n})$ and $\frac{1}{n}$ tends to $0$, since $n$ tends to infinity, so the limit becomes ...
2
votes
6answers
5k views

Which Mathematical Analysis I Book or Textbook Is The Best?

I'm in search of a mathematical analysis text that covers at least the same material as Walter Rudin's Principles of ... but does so in much more detail, without relegating the important results to ...
2
votes
2answers
99 views

Does this hold for product Lebesgue measure?

Let $\mathscr M_n$ denote the set of lebesgue measurable subsets of $\Bbb R^n$. Given a set $K\subset \Bbb R^{n+m}$ with the property: $\forall x\in \Bbb R^n: K_x:=\{y\in\Bbb R^m\mid(x,y)\in K\} \in ...
3
votes
1answer
124 views

How to find all polynomials with rational coefficients s.t $\forall r\notin\mathbb Q :f(r)\notin\mathbb Q$

How to find all polynomials with rational coefficients$f(x)=a_nx^n+\cdots+a_1x+a_0$, $a_i\in \mathbb Q$, such that $$\forall r\in\mathbb R\setminus\mathbb Q,\quad f(r)\in\mathbb R\setminus\mathbb Q.$$ ...
0
votes
0answers
56 views

For what value of $p$ does this hold?

For which values of $p>0$ does the inequality $|x+y|^p\le |x|^p+|y|^p$. I am thinking convexity but I am not sure. thanks
1
vote
0answers
40 views

Bounds on coefficients of factors of a multivariate polynomial

If I have a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that I can quickly find such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote ...
1
vote
2answers
81 views

frontier of class $C^{1}$.

Studying the Divergence Theorem (Gauss theorem), found the definition of frontier of class $C^{1}$. Which means? That is, the one which is a set with boundary of class $C^{1}$? Can give reference ...
3
votes
1answer
213 views

$d_2=\frac{|x-y|}{1+|x-y|}$ is a metric

I am pretty sure that this is such a stupid, stupid question, but how do you prove that $d_2={|x-y|\over {1+|x-y|}}$ satisfying the third condition to be a metric, which is the triangle inequality. ...
1
vote
2answers
309 views

Product of two Riemann integrable functions

I want to show the following inequality is true: $|\int_{a}^{b}fg|^2\leq \int_{a}^{b}f^2\int_{a}^{b}g^2$. My first thought was to use the tagged partition definition of a Riemann integral combined ...
3
votes
2answers
333 views

Showing a function is not Riemann integrable

Let $m,n \in \mathbb{Z_{+}}$ and let $f(x)=\begin{cases} x^m+x^n \text{ if } x \in [0,1]\cap\mathbb{Q}\\ 0 \text{ if } x \in [0,1] \setminus \mathbb{Q} \end{cases}$. I thought of a similar function ...
3
votes
0answers
76 views

Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
1
vote
0answers
57 views

How to observe such an inequality?

Yesterday somebody post a problem on my college bbs: Let $f(z)=1+\sum\limits_{n=1}^{+\infty}a_{n}z^{n}$ be a analytic function with positive real part on the unit disc $D=\{z:|z|<1\}$, please show ...
3
votes
1answer
440 views

Prove partial derivatives of uniformly convergent harmonic functions converge to the partial derivative of the limit of the sequence.

I think the title says it all. If you have a sequence of harmonic functions from a bounded complex domain to the real numbers, show that on a subset at a positive distance from the boundary of the ...
0
votes
0answers
70 views

Union of Countable Sets is Countable [duplicate]

Let $\{E_n\}$, $n = 1, 2, 3, \ldots$, be a sequence of countable sets, and put $S = \displaystyle \bigcup_{n=1}^{\infty} E_n$. Prove that $S$ is countable.
4
votes
1answer
391 views

proof of Poisson formula by T. Tao

I do not understand one thing in an article on the blog of Terence Tao: For instance, restricting a function $f: G \rightarrow \mathbb{C}$ to a subgroup $H$ causes the Fourier transform $\hat f$ ...
3
votes
4answers
195 views

how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?

Let$a_n,b_n\in\mathbb R$ and $(a_n+b_n)b_n\neq 0\quad \forall n\in \mathbb{N}$. The series $\sum_{n=1}^\infty\frac{a_n}{b_n} $ and $\sum_{n=1}^\infty(\frac{a_n}{b_n})^2 $ are convergent. How to prove ...
7
votes
2answers
511 views

Accumulation Point at Infinity?

This is a follow up question to the one I posted here. So my new question involves accumulation points and infinity. Can infinity be considered an accumulation point? For example consider the ...
2
votes
1answer
304 views

On Proof Techniques

Clearly, there are standard ways of proving general theorems without regard to the actual subject matter. These include, but are not limited to, proof by contrapositive, proof by contradiction, and ...
3
votes
1answer
225 views

Proving continuity of an integral

I have the following function: $$I_n(a)=\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(ax)dx$$ where $\operatorname{sech}(x)=\frac{2}{e^x+e^{-x}}$ is the hyperbolic secant. Clearly, the ...
1
vote
1answer
74 views

Proving some basic facts about integrals

Let $f$, $g$ be Riemann integrable functions on the interval $[a,b]$, that is $f,g \in \mathscr{R}([a,b])$. (i) $\int_{a}^{b} (cf+g)^2\geq 0$ for all $c \in \mathbb{R}$. (ii) $2|\int_{a}^{b}fg|\leq ...
0
votes
3answers
124 views

Given $n+1\mid2\sum_{k=1}^{n}{a_k}$, find $a_k$.

Let $m$ be a positive integer. There are only 2 finite sequences of positive integers like $a_1,a_2,...,a_m$ such that $$(\forall n \leq m)\left(n+1\mid2\sum_{k=1}^{n}{a_k}, \quad a_n\in [1,m],\quad ...
1
vote
1answer
101 views

Why this theorem implies the equicontinuity of the second derivative?

In the book of Gilbar Trudinger (Elliptic partial differential equations of second order), the theorem 4.6 (page $60$) give the following estimate: $$|u|'_{2,\alpha;B_1}\leq ...
0
votes
1answer
258 views

Is the integral of a smooth function continuous?

Suppose I have a function $f(a,x):\mathbb{R}^2\rightarrow\mathbb{R}$ that is smooth (i.e. infinitely differentiable) over its entire domain $\mathbb{R}^2$. Let $I(a)=\int_{-\infty}^{\infty}f(a,x)dx$. ...