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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4answers
55 views

$\sum_{n=1}^\infty \frac{n+1}{\sqrt{n^3+1}}$convergent/divergent?

Please could someone help prove $$\sum_{n=1}^\infty \frac{n+1}{\sqrt{n^3+1}}$$ converges/diverges? Thank you.
1
vote
0answers
93 views

Find the principal part of $\tan z + \tan(z-a)$

How can I find the principal part of $\tan z + \tan(z-a)$ ? Do I have to find the Laurent Expansion of each function, then do the sum and after that find the principal part? Finding the Laurent ...
2
votes
1answer
116 views

How do I prove this function is not continuous?

Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$. The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
5
votes
0answers
278 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
0
votes
3answers
96 views

How can we prove that $\left | (a-b)^{2}+ab \right | \geq \left | ab \right |$?

I have the multivariable function : $$f_{\alpha}(x,y)=\frac{\left | xy \right |^{\alpha}}{x^{2}-xy+y^2}$$ I think that an upper bound in $(0,0)$ for $\alpha > 1$ is : $$\left | f_{\alpha}(x,y) ...
5
votes
3answers
318 views

Differentiability of the distance function

Suppose that $d:X \times X \to \mathbb{R}$ is a geodesic distance function on a smooth Riemannian manifold $X$ ($d$ is determined by metric tensor) and $x \in X$ is fixed. What can be said about the ...
1
vote
0answers
36 views

A branch of logarithm $L$ such that $f(z)=L(z+i-2)$ is analytic at $-i$

How do I find a branch of logarithm such that $f(z)=L(z+i-2)$ is analytic at $-i$ and $f(-i)=log2+2\pi i$? If we take $g(z)=z+i-2$ this is analytic at $-i$. Also, $f(-i)=L(-2)=log2+2\pi i$. I do not ...
2
votes
1answer
164 views

Continuous function on $\mathbb{R}^{n}$ preserving compactness - some clarification

My professor went over a proof of the following in class: Suppose $A \in \mathbb{R}^{n}$ is compact and $f:A \rightarrow \mathbb{R}^{n}$ is continuous. Then $f(A)$ is compact. The proof ...
0
votes
1answer
42 views

Taylor-expansion for a limit expression on $\mathbb{R}\setminus\mathbb{Z}$

Say $$f(x)=\frac{\pi^2}{\sin^2(\pi{}x)}-\sum\limits_{k\in{\mathbb{Z}}}\frac{1}{(x-k)^2}.$$ And $f$ is defined on $\mathbb{R}\setminus\mathbb{Z}$. Prove by the Taylor-expansion of $x\mapsto ...
0
votes
3answers
293 views

How to visualize a convex set is connected but not vice-versa.

Can anyone explain this graphically/intutively to me: A convex set is always a connected set while the converse is not true.... (A convex set is a set in which every element $a,b$ belonging to ...
2
votes
1answer
51 views

terminology relating to o(1)

If someone says, for example, "I have an algorithm that runs in time $n^2+\varepsilon$ for any constant $\varepsilon>0$", the interpretation for this statement seems to be that for any constant ...
0
votes
3answers
38 views

Proving $x^2 < y^2$ by means of the Ordering Axioms [closed]

How do I prove $x^2 < y^2$, if $0 \le x < y$ with the ordering axioms? thanks!
1
vote
1answer
36 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
0
votes
1answer
31 views

Proving for each seperatble hilbert space exist complete sequence

Let $H$ be a separable Hilbert Space. Prove that exists orthonormal complete sequence and give example for one non-orthonormal sequence. I thought taking orthonormal basis for $H$ denoted by ...
1
vote
1answer
96 views

Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence

Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that ...
1
vote
0answers
64 views

Cantor-Lebesgue Functions

Show that there is a continuous, strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero. Is it enough to prove that a strictly increasing ...
0
votes
2answers
54 views

In a metric space with a countable base, how does every open cover has a countable subcover?

Let $X$ be a mertic space, and let $\{V_{\alpha}\}$ be a collection open subsets of $X$ such that, for every $x \in X$ and for every open set $G \subset X$ with $x\in G$, there is some $V_\alpha$ such ...
0
votes
1answer
90 views

Is this a Lipschitz continuous function?

Is $$\frac{xy}{1 + x^2 + y^2}$$ Lipschitz continuous on $x^2 + y^2 \le 4$? I've tried using Cauchy-Schwarz intequality but got nothing. I also tried to find out whether $xy$ is Lipschitz but failed ...
2
votes
2answers
107 views

For $A=\{\sin (2n\pi/7) \mid n \in \mathbb{N}\},$ how do I find $\sup(A)$ and $\min(A)$?

I'm kinda new at this and I know what $\sup$ and $\min$ mean, but the problem is when calculating them like the example above. Can you enlighten me please?
2
votes
4answers
301 views

Why do we need min to choose $\delta$?

On this thread: Problem: The person uses $\delta = \text {min} (\frac{\epsilon}{2}, \frac{1}{2})$ why do we need the min function to determine $\delta$? Thanks!
2
votes
1answer
53 views

Isn't that proof going the wrong way?

I'm currently working on the very well written book Understanding Analysis, by Stephen Abbott. But I found a proof that looks wrong, I think that it going the wrong way (showing that A $\implies$ B ...
0
votes
0answers
52 views

Estimates on solution of pde

Let's consider the following pde in $\mathbb{R}^n$ $$\partial_t u=(i+\varepsilon)\Delta u,\,\,\,u(0,x)=u_0(x)$$ How to get the following estimate for its solution? $$\Vert u\Vert_2\leq C_\varepsilon ...
0
votes
2answers
30 views

$x/|x|$ question about division

What is $\frac{x}{|x|}$ can it be simplified? Because look at this. $\frac{r\cosh(x)}{\sqrt{\cosh^2(x)}} = \frac{r\cosh(x)}{|\cosh(x)|}$ How do you do this?
1
vote
1answer
93 views

Some problems from section 4 of Munkres

I'm right now covering Section 4 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problems in the exercise set after Section 4: Problem 8(c): Show that given $a$ with ...
2
votes
0answers
62 views

Fundamental solution of Poisson equation in the Hyperbolic Plane

If we consider the Poisson's equation $$ -\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n, $$ we can construct the fundamental solution $$ u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy, $$ where $\Gamma$ is ...
2
votes
2answers
73 views

I would like prove a result in integration

I would like prove this result $$\int_0^1 \frac{\left(\log (1+x)\right)^2}{x}\mathrm dx=\frac{\zeta(3)}{4}$$
0
votes
1answer
95 views

A proof I don't understand in analysis

I am reading a book in analysis it is proving the Ratio test, but their is some step I don't understand or I am not entirely sure of how they got that result. (Ratio Test) Let $\sum a_n$ be a series ...
0
votes
1answer
34 views

Connections between two definitions of locally Lipschitz mapping

Let's consider two following definitions of locally Lipschitz mapping. Let $f: D\subset \mathbb R^n \rightarrow \mathbb R$. We say that $f$ is localy Lipschitz in 1. sense, if for each $a\in D$ ...
1
vote
3answers
80 views

Prove this limit (formally)

As I was coursing through Spivak's calculus, more like analysis; I found an interesting, questionable example. Let $\frac{p}{q}$ be in its lowest terms; $p$ and $q$ are integers with no common ...
2
votes
1answer
23 views

uniform convergence of $u_\varepsilon(x)=-\varepsilon\log\left(\frac{\cosh(\frac{x}{\varepsilon})}{\cosh(\frac{1}{\varepsilon})}\right)$

I want to prove that the sequence $$u_\varepsilon(x)=-\varepsilon\log\left(\frac{\cosh(\frac{x}{\varepsilon})}{\cosh(\frac{1}{\varepsilon})}\right)$$ converges uniformly to $u(x)=1-|x|$ for ...
1
vote
1answer
74 views

Relationship between the Hausdorff dimension and the Box-counting dimension

In Fractal Geometry by Falconer the author writes: If $1<\mathcal H^s(F)=\lim_{\delta\to0}\mathcal H_\delta^s(F)$ then $\log N_\delta(F)+s\log\delta>0$ if $\delta$ is sufficiently small. ...
10
votes
1answer
245 views

Can the set of computable numbers be used as a theoretical basis for calculus?

I recall from my Real Analysis course that the rational numbers $\mathbb{Q}$ are not suitable for doing calculus, and I believe the reason was that $\mathbb{Q}$ does not possess the least-upper-bound ...
0
votes
1answer
38 views

Need help with a basic proof using Paeno Axioms showing that a recursively defined function is one to one

Assume $(N, 0_N, ++^N) and (N˜, 0_{N˜} , ++^{N˜} )$ are two systems satisfying the Peano axioms. Define the function $T : N → N˜$ by $T(0_N) = 0_{N˜}$ and $T(n++^N) = T(n)++^{N˜}$ Prove that ...
0
votes
1answer
36 views

Continuous on an interval implies continuous on every subinterval?

Let $f:\mathcal{I} \longrightarrow \mathbb{R}$ be a continuous function. Is this function continuous on every subinterval of $\mathcal{I}$? If yes, prove it. If no, give a counterexample. I have ...
2
votes
1answer
44 views

How to prove that if $\|f_n-f\|_p \rightarrow 0$ then $\|F_n-F\|_p\rightarrow 0$

Let $f_n \in C_c^\infty(0,\infty)$ for $n\in \mathbb N$, $f: (0,\infty) \in L^p(0,\infty)$, where $1<p<\infty$ and $\|f_n-f\|_p \rightarrow 0 $ as $n\rightarrow \infty$. We define $$ ...
0
votes
2answers
90 views

Proof of a property of directional derivative

I am stuck with the proof of the following proposition. I am given that the directional derivative of f exists at a with respect to the vector u, and I should prove that f'(a,cu)=cf'(a,u) I tried to ...
4
votes
2answers
57 views

What is a solution of such equation concerning the arithmetic and integral means?

Let $f:[a,b] \rightarrow \mathbb R$ be integrable and satisfies $$ f\left(\frac{x+y}{2}\right)=\frac{1}{y-x} \int_x^y f(t)dt $$ for all $x \neq y$, $x,y \in [a,b]$. What about $f$? Is it affine ...
2
votes
1answer
52 views

Continuous linear functional and weak convergence

I have a question about a continuous linear functional. $T>0$ : fix. $C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$ $C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$ Then ...
0
votes
1answer
52 views

Bi-Lipschitz invariance of the box-counting dimension.

I would like to prove that the box counting dimension is invariant under a bi-Lipschitz transformation. We have that $f$ is bi-Lipschitz if there exists $c_1, c_2$ such that $0 < c_1 \leq c_2 < ...
1
vote
1answer
30 views

Compact Operator Inversion

Let I be a positive compact in $\mathscr{B}(\mathscr{H})$ (where $\mathscr{H}$ is some Hilbert space) then $I$ can be written (uniquely) as $A^2=I$ for some $A \in \mathscr{B}(\mathscr{H})$. My ...
1
vote
0answers
48 views

question about Skorokhod distance

Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$ ...
0
votes
2answers
42 views

How is this boundary condition $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$ for a PDE called?

If you have a diffusion equation $\partial_t f(x,t) = \partial_x^2 f(x,t) $, where $(x,t) \in [0,a] \times \mathbb{R}$ and then you say $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$, how do you call ...
0
votes
2answers
162 views

Spivak “min” notation confusion

Spivak uses a notation: min$(1, \frac{\epsilon}{2|a| + 1})$ What does he mean by this notation? especially by "min"??
2
votes
2answers
54 views

What is the largest open set $\frac{1}{\cos z-2i}$ is analytic in?

This is a very interesting question that I came across and have never solved any question of this sort. How do I find the largest open set on which $\frac{1}{cosz-2i}$ is analytic? Do I find the set ...
1
vote
2answers
66 views

Derivative of matrix inversion function?

Let's say I have a function $f$ which maps any invertible $n\times n$ matrix to its inverse. How do I calculate the derivative of this function?
1
vote
1answer
51 views

Second order linear ODE $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$

I have $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$ How do I solve this? What have I tried? $1)$ Coupled system: $\begin{pmatrix}y_1^{\prime} \\ ...
0
votes
1answer
272 views

Prove that lim sup $a_n$ $\leq$ lim sup $b_n$.

Let $a_n$, $b_n$ be bounded sequences of real numbers and suppose that there exists N so that $a_n$ $\leq$ $b_n$ for all n $\geq$ N. Prove that lim sup $a_n$ $\leq$ lim sup $b_n$. I can see why ...
1
vote
1answer
72 views

How find this sum $I\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)}=1-\frac{1}{2}\ln{(2\pi)}$

Question: show that $$I=\sum_{k=1}^{\infty}\dfrac{B_{2k}}{2k(2k-1)}=1-\dfrac{1}{2}\ln{(2\pi)}$$ where $B_{n}$ is Bernoulli number:Bernoulli number I think we can ...
2
votes
1answer
78 views

Distributions (Generalized Functions)

Why is a distribution defined in terms of the inequality $$ |\langle\Gamma, \psi\rangle| \leq C \sum_{|\alpha| \leq N} \sup_{x \in S} | \partial^\alpha \psi |$$ for all $\psi \in C^\infty_c ...
1
vote
1answer
46 views

Dirichlet characters - proof in a book

I found the following in a book and don't understand. Let $\chi$ denote a non-principal character modulo $q$ and $S(x)=\sum_{n\leq x}\chi (n)$. Then $\sum_{m>y} \frac{\chi(m)}{m} = \int_y^{\infty ...