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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3
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1answer
34 views

Triangle inequality and homomorphisms

Here is my situation: I have two homomorphisms $f$ and $g$ from a group $A$ into the complex numbers $\mathbb{C}$. I know that they are 'close' on a subset $B \subseteq A$. More formally there is an $\...
4
votes
1answer
103 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
1
vote
1answer
61 views

Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
1
vote
1answer
59 views

nth term derivative with f(0) plugged in

Question: compute the sixth derivative of $\frac{\cos{(5x^2)}-1}{x^2}$ and plug in zero to the derivative. What is the answer? I keep concluding that this question should be 892857 when I plug in 0 ...
4
votes
1answer
140 views

Riemann Sums as in Königsberger Analysis 1

Intro: I must take a small detour here which is only relevant if you do not know the book itself and care about my background. I am working with Königsberger Analysis I (can be found on Springerlink). ...
2
votes
1answer
48 views

Clarification about a basic proposition about measurable functions

I am making my way through "Linear Functional Analysis" by Bryan P.Rynne and Martin A.Youngson (second edition). Given a measure space $(X,\Sigma ,\mu )$ we define a function $f$ to be measurable if ...
2
votes
2answers
179 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
5
votes
1answer
102 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
1
vote
1answer
43 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
2
votes
1answer
19 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
2
votes
1answer
101 views

Positive and negative Parts

we denote by $u^+=\max(u,0)$ and $u^-=\max(-u,0)$ the positive and the negative parts of $u$ we have that $u=u^+-u^-$ my question is : what is $u'$ using $u^+$ and $u^-$ ? and what is $\int_{\Omega}...
0
votes
1answer
60 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, (1\...
1
vote
1answer
48 views

Converge of an inversion to a mirrorring

I want to ask something about a mirroring and a inversion in $\mathbb{R}^n$. An inversion in a sphere with center $m$ and radius $\rho$ can be written as $$ v \ \longmapsto \ \frac{\rho^2}{\|v-m\|^2}...
1
vote
1answer
48 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k &...
0
votes
1answer
43 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
0
votes
1answer
77 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
3
votes
2answers
66 views

integration of a series in $x\in(0,1)$

Prove that if $p>0$, $$ \int_0^1 \frac{x^{p-1}}{1-x}\log\left(\frac{1}{x}\right)dx=\frac{1}{p^2}+\frac{1}{(p+1)^2}+\dots$$
1
vote
0answers
54 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
1
vote
1answer
29 views

Question about convergence in $H^1_0$

Please how to prove that if $u_n\rightarrow u$ on $H^1_0$ we have that $||u_n||\rightarrow ||u||$ ? Please i need your help Thank you
1
vote
1answer
68 views

$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in C_{c}^{\infty}(\mathbb R), \|f\|_{L^{\infty}(\mathbb R)}=1\}} ?$

I learn the following from the book: Fact: If $g\in L^{1}(\mathbb R),$ then $$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in L^{\infty}(\mathbb R), \|f\|_{L^{\infty}}}=1\}.$$ We put $...
0
votes
1answer
63 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
1
vote
1answer
37 views

Small question about convergence

I have a small question: if i have that $$\int_0^{+\infty}p(t)|u'_n(t)-u'(t)|^2dt\rightarrow 0$$ is it true that $$\int_0^{+\infty} p(t)|u'_n(t)|^2 dt\rightarrow \int_0^{+\infty} p(t)|u'(t)|^2 dt $$ ...
0
votes
0answers
61 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} \|\...
2
votes
2answers
69 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
3
votes
2answers
94 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields $$f(z)=\underbrace{\frac{1}{z+4}}_{=:...
1
vote
1answer
139 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ $$||u||^2=\int_0^...
2
votes
1answer
35 views

How to proof the limit is convergent for arbitrary initial state?

$\{a_n\},\{b_n\},\{c_n\},\{d_n\}$ is series. And $d_n=c_{n-1},c_n=b_{n-1},b_n=a_{n-1},a_n=b_{n-1}+c_{n-1}$ how to proof for any $a_0,b_0,c_0,d_0$ belong to $Z^+$, $\lim\limits_{n\rightarrow \infty} \...
1
vote
1answer
43 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
3
votes
1answer
56 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
2
votes
1answer
144 views

Can't understand a step in the advanced calculus book by thomas P. Dence

On page 9 of the Advanced calculus book by Thomas P. Dence he defines the set $S_1 = \{x\in \Bbb Q: x\geq 0, x^2 \leq 5\}$ and he said that the supremum $U$ is less than $u=3$ and that $U$ is ...
0
votes
0answers
48 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
4
votes
2answers
297 views

Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
3
votes
1answer
104 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
0
votes
1answer
68 views

Find a weak derivative of this function

Let $B(0,1)$ be the open unit ball in Euclidean space $\mathbb R^2$ and $(a_n)_{n=1}^\infty$ be a dense subset of $B(0,1)$. I wish to show for fixed $s\in (0,1)$ the function $$ f(x)=\sum_{n=1}^\...
1
vote
1answer
80 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists (z_k)_{k\in\...
0
votes
1answer
44 views

Maximum value of function involving factorials

Define $$g_{(k,j)} = \frac{a^{n-k}b^k(k+n)!x^{k+n-j}}{k!(n-k)!(k+n-j)!}$$, where $n,k,j \in \Bbb{N}$ are fixed such that $(0 \leq x \leq a/b ),(b<a),(0 \leq k \leq n ),(2 \leq j \leq 2n),(0 \leq k+...
2
votes
1answer
97 views

Generalizing the Monotone Subsequence theorem

In proving the Bolzaono-Weierstrass theorem, one proves the lemma that every infinite real sequence has a(n infinite) monotone subsequence. In all of the proofs I've seen so far, this is done by ...
0
votes
1answer
16 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in (-\...
-1
votes
1answer
32 views

Part limit in Complex Analysis

Can somebody explain to me why this limit: $lim(n(\sqrt[n]{r} - 1))$ converges to $logr$ when $n \rightarrow \infty$? Thanks a lot! This was a part of a limit in Complex Analysis asking to show ...
4
votes
2answers
70 views

Show that f is measurable

Let $U$ be a open Set of $\mathbb{R} \times [0,\infty]$ and let f be defined as $$f: \mathbb{R}\mapsto [0,\infty], \quad f(x) := \max\{0,\sup\{y| (x,y) \in U\}\} $$ How can I show that $f$ is ...
1
vote
1answer
77 views

A direct application of inverse function theorem

Let $f:U\longrightarrow \mathbb{R}^n$ a function with $U\subset \mathbb{R}^n$ open, $f$ injective of class $C^1$ (i.e. continuous with the first derivate continuous) such that $\forall x\in U$ the ...
2
votes
3answers
196 views

Intuitive Numerical Analysis Texts

Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement ...
4
votes
2answers
841 views

Show a function is monotonically decreasing.

Show that $f(x)=\dfrac{\sin x}{x}$ is monotonically decreasing on $[0,\frac{\pi}{2}]$ I'm trying to show that $f'(x)\leq0$ to show it's monotonically decreasing. So $f'(x)=\dfrac{x\cos x-\sin x}{x^2}...
4
votes
1answer
99 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 \int_{x^2+...
1
vote
2answers
2k views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty)$ be a nonnegative, ...
2
votes
1answer
92 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
1answer
20 views

Does convergence of $S_{n!}=\sum{1}^{n!}a_{k}$ implies $a_{k}$ approaches to zero?

**Let $\{{a_n}\}$ be a sequence . Define $ S_{n}=\sum_{1}^{n}a_{k}$. Does convergence of $\{{S_{n!}}\}$ implies $\lim{a_{n}}=0$ as $n\rightarrow\infty$. **
1
vote
0answers
81 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in L^{\infty}(...
3
votes
6answers
228 views

Constructive proof of Euler's formula

In most textbooks on the subject I have seen, Euler's formula (by which I mean $e^{ix}=\cos(x)+i\sin(x)$) is proved by applying either differential equations or the power series of sine and cosine. ...
2
votes
2answers
97 views

How find the function $f(x)$ such $\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$

let $f(x)$ is Continuous function on $[0,\pi]$,and for infinite positive integer $n$ such $$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$$ Find the $f(x)$? I think the answer is $f(x)=c$?,But maybe have ...