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
84 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
9
votes
1answer
153 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
2
votes
1answer
52 views

If $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is $\mathcal{C}^1$, then $f$ is not one-to-one.

I tried to use the contour line and the problem is equivalent to show that the contour line of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is not a single point. I also thought about the Mean ...
3
votes
1answer
78 views

Showing irrationality of $\zeta(k)$ for some $k$ without calculating the value.

For $s\in (1,\infty)$ let $\zeta(s):=\sum_{n=1}^\infty \dfrac 1{n^s}$. Is there a way to show that $\zeta(2k)$ is irrational for some integer $k\geq 1$ without finding explicit formulae?
1
vote
3answers
99 views

prove when $r \geq t $ then $ x^{r} \geq x^{t} $

when $ r,t \in \mathbb{Q} , x \geq 1 $ and $ r\geq t $, prove $ x^{r} \geq x^{t} $. i have tried so much but i can't prove it. :( My attempt : i can derive from it that this proposition is equal ...
0
votes
1answer
19 views

Prove that $\sum_{x\in I}\delta_x$ diverges if $\delta_x>0$ and $I$ is more then countable.

Take $F:\Bbb R\to\Bbb R$ s.t. $F$ is continous from the right, not decreasing, $\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to+\infty}F(x)=1$. Call $I:=\{x\in\Bbb R\;:\;F\;\; \mbox{is not continous ...
6
votes
2answers
88 views

How prove this $\frac{af(a)+bf(b)}{a+b}\ge f(a+b)$

Assume that $f(x)$ has two derivatives on $(0,2)$ and $0<a<b<a+b<2$. I have to prove that, if $f(a)\ge f(a+b)$ and $f''(x)\le 0$, then: $$\dfrac{af(a)+bf(b)}{a+b}\ge f(a+b).$$ I ...
0
votes
2answers
1k views

how to show that a function is unbounded?

How to prove that the function $f:(0,2)\to\mathbb{R}, f(x)=\frac{1}{x}$ is unbounded. I know for a function is unbounded if: $\forall M>0 \exists x\text{ such that }|f(x)|>M$
2
votes
1answer
37 views

Does $g'$ need to be continuous for $g(x_0) = 0$, $g'(x_0) \neq 0$ to imply $g$ changes sign in a neighborhood of $x_0$

The following theorem holds: Theorem: Let $g:\mathcal{A} \rightarrow \mathbb{R}$ be differentiable and let $x_0 \in \mathcal{A} $. If $g(x_0)=0, \; g'(x_0)\neq 0$ then $g$ changes sign at a ...
0
votes
1answer
35 views

Is $T:(x,y)\mapsto(x+\alpha, y+x)$ mod $1$, expansive on $\mathbb{R}^2 / \mathbb{Z}^2$?

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha, x+y\right) \mod 1 $$ One ...
1
vote
1answer
91 views

How prove this limits $\lim_{n\to\infty}\int_{0}^{1}g(x)f'(nx)dx=0$

let $f'(x)$ be continuation on $R$, and $f(x)=f(x+2\pi),\forall x\in R$, and such $|f'(x)|\le 1$,Now define $g(x)$,such for any $x_{1},x_{2}\in [0,1]$, we have $$|g(x_{1})-g(x_{2})|\le ...
0
votes
0answers
28 views

Divergence proof problem in introductory analysis text.

The problem is this: Show that if $a_n > 0$ and $\lim_{n\to \infty} na_n = L$ with $L \neq 0$, then the series $\sum a_n$ diverges. (from Abbott's Understanding Analysis, p. 68). I want ...
2
votes
2answers
50 views

Topology on k((t))

$k((t)):=\lbrace (a_i)_{i \in \mathbb{Z}}, a_i \in k,\exists \ N \in \mathbb{Z} \ s.t \ \forall \ i<N, a_i=0\rbrace$ where $k$ is a field of char zero. We define componentwise addition and ...
2
votes
3answers
161 views

Why is a norm a continuous function? (Question about existing proof)

I'm trying to follow the proof given in this answer: http://math.stackexchange.com/a/265595/188401 I understand the proof in general, but I have a question. It's mentioned that "In this case it ...
6
votes
1answer
77 views

How derivative relates to roots of original function

Assume $f$ is differentiable on $\mathbb{R}$. Show that for any $ k \in \mathbb{R}$, $f' + kf$ has a root between any two distinct roots of $f$. I am completely stumped on this. What are some good ...
1
vote
6answers
159 views

Does $y_n=\frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n}$ converge or diverge?

I have to show whether $y_n=\frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n}$ is convergent or divergent. I tried using the squeeze theorem to prove it was convergent. So what I did was bound ${y_n}$ in ...
1
vote
1answer
198 views

Covariant derivative and geodesic

Let $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a surface patch. Then if we have two vector fields $$X = \sum_i \xi^i \frac{\partial f}{\partial u^i}$$ and $$Y = \sum_i \eta^i ...
1
vote
1answer
77 views

Uniform Convergence proof and limit inside integral

Suppose we have: $$f_n(x) = ie^{ine^{ix}}$$ We are to evaluate: $$\lim_{n \to \infty} \int_{0}^{\pi} ie^{ine^{ix}} dx$$ We must first prove that $f_n(x) \to f(x)$ uniform convergence. I am not ...
0
votes
3answers
102 views

Is being a Cauchy sequence equivalent to $ \lim_{n\to+\infty}d(x_{n+k},x_n)=0$ for every $k$?

Is this statement true? In a metric spase $(E,d)$, a sequence $(x_n)$ is Cauchy if and only if $ \forall k\in \mathbb{N}, \lim_{n\rightarrow+\infty}d(x_{n+k},x_n)=0$ I proved that ...
1
vote
3answers
73 views

Real and imaginary part of $ (1-i\sqrt{3})^6$

i am a bit stuck here. As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$ I already found out that the ...
6
votes
1answer
120 views

Evaluate $\int_{-\infty}^{\infty} \frac{\log(1+x^2) dx}{1+x^2}$ Using Complex Analysis

Evaluate: $$\int_{-\infty}^{\infty} \frac{\log(1+x^2) dx}{1+x^2}$$ Using complex analysis, contour integration. This function has no poles at all. Try the contour $C$ Obviously, ...
0
votes
2answers
66 views

Dominated convergence theorem on $e^{ix}$

I am considering first: $$\lim_{n \to 0} \int_{0}^{\pi} e^{ine^{ix}} dx$$ To bring the limit inside I need to apply the dominated convergence theorem. Keep in mind I have no knowledge of measure ...
1
vote
1answer
59 views

Convergence of series/absolute convergence

Let $y_n$ be a sequence of real numbers such that for all sequences of real numbers $x_n$ with $\lim x_n =0$ the series $\displaystyle \sum_{n=1}^{\infty} x_n y_n $ converges. Prove that ...
0
votes
3answers
80 views

How to prove that $|f(b)-f(a)|\le\frac{1}{n}(b-a)^2$

for any real numbers $x,y$ we have $$f(y)-f(x)\le (y-x)^2$$ show that: for any postive integer $n$,and any real numbers $a,b$,we have $$|f(b)-f(a)|\le\dfrac{1}{n}(b-a)^2$$ My partial ...
1
vote
1answer
51 views

How to solve this equality? [duplicate]

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF How 1+2+3+ ... = - 1/12 ? Why? Shouldn't the result be infinite?
2
votes
1answer
102 views

Is it always possible to use Chasles to decompose an integral?

Well, I'm in "classe préparatoire" and I always learn that if f is a continuous fonction integrable on [a,b] and if c is in [a,b] then with Chasles relation we have : $$ \int_a^b f(x) dx = \int_a^c ...
0
votes
0answers
45 views

Prime Space - straight lines only connect primes?

I was reading a very obscure article printed out at my university's library, and there was a topic which I wish to discuss a bit further. That is, the author defined a space, namely $\mathbb{P}$, ...
3
votes
2answers
121 views

Can any metric space be completed?

Completion defined in Real Analysis, Carothers, 1ed has been captured below. Can any metric space be completed?
2
votes
1answer
17 views

A property of a solution of a differential problem.

Let $f \in L^2(0,1)$ such that $f(x)=f(-x)$ a.e. in $(-1,1)$ and let $u$ be the solution of the problem $$ -u''(x)+u(x)=f(x) \,\,\,\, x \in (-1,1)$$ with the condition $$ u(-1)=u(1)=0 $$ Can I ...
1
vote
0answers
57 views

Properties of solutions of system of integral equation.

Assume $g:[0,\infty) \to \mathbb R$ to be continuous and $$\int_{0}^{\infty} s|g(s)| \,\mathbb ds< \infty .$$ I want to find $\alpha>0$ such that the system of integral equations ...
17
votes
1answer
597 views

Integral $\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}$

Please help me prove the following identity: $$\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}\quad a>0$$ This integral is from ...
0
votes
2answers
150 views

Is there only one continuous-everywhere non-differentiable funtion?

I'm reading Landau's: Differential and Integral Calculus. On theorem 100, he states: There is an everywhere-continuous nowhere-differentiable function. I've read somewhere that these patological ...
0
votes
1answer
118 views

Homeomorphism between two locally compact spaces

Suppose $X_1$ and $X_2$ are two locally compact spaces. Define $\phi:X_1\to X_2$. Suppose $\phi$ is bijective and continuous. I know that if $X_1$ is compact, I can conclude $\phi$ is a homeomorphism. ...
1
vote
0answers
30 views

$ \|u(x)+u(x)-x +o(\|x\|^p)\|<r $?

Set $B_r=\{ x\in \mathbb{R}^n : \|x-0\|< r \}$ for any $r>0$. Let $C^p_0(B_r,B_r)$ the set of all smooth functions $u:B_r \to B_r$ of class $C^p$ such that $u(0)=0$. I would like to prove the ...
1
vote
0answers
85 views

If $\langle\xi\rangle=(1+|\xi|^2)^{\frac 12}$, is the inverse fourier transform of $\frac{\langle\xi\rangle^{-n}}{1+\log{\langle\xi\rangle}}$ bounded?

Let $\langle\xi\rangle=(1+|\xi|^2)^{\frac 12}$. Is $\mathcal{F}^{-1}(\frac{\langle\xi\rangle^{-n}}{1+\log{\langle\xi\rangle}})$ a bounded function? $\mathcal{F}^{-1}$ denotes the inverse Fourier ...
5
votes
3answers
1k views

Continuous function with local maxima everywhere but no global maxima

Can there be such a function: $f \colon \mathbb R \to \mathbb R$ is continuous and non-constant. It has a local maxima everywhere, i.e., for all $x \in \mathbb R$ there is some $\delta_x>0$ such ...
6
votes
1answer
107 views

Is continuous $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$?

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous. Is $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$? What metric spaces we can use instead of $\mathbb{R}$? I guess we ...
4
votes
0answers
66 views

Interchangeability of the malliavin derivative with a lebesgue integral

I was curious to know the most general conditions under which a malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a lebesgue integral? I was ...
1
vote
1answer
115 views

Is the condition of continuity for the differentable functions necessary in Looman-Menchoff theorem?

Looman-Menchoff theorem states that a continuous complex-valued function $f(z)=u(x,y)+iv(x,y)$ defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann ...
2
votes
2answers
262 views

Why do we have the notions of both 'norm' and 'metric'?

In the case of vectors in euclidean space, for instance, we can express one in terms of the other--i.e. length is distance from zero, distance is the length of the vector difference. Does this break ...
0
votes
0answers
32 views

Proving the solutions to an Equation

Consider: I am sort of confused. Without using the intermediate value theorem, directly. I suppose indirect use if fine. $1 + x^2 + \sin^2(x) > 0$ for $x \in \mathbb{R}$ Let the $LHS = I$ $$I ...
0
votes
2answers
140 views

Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
3
votes
1answer
104 views

Mathematical Analysis: What is the Velocity of the Falling object through each point in its path?

I have been working through the book called "Mathematics" written by A.D. Aleksandrov, A.n. Kolmogorov and M.A. Lavrent'ev recently and have had some difficulty with understanding Examples given by ...
0
votes
3answers
52 views

Is this a valid sum formula for rational functions?

Consider: $$\frac{a(x)}{\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1}x^{2n-2}}$$ Is the expressions above equivalent to: $$\sum_{n=1}^{\infty} \frac{a(x)}{(-1)^{n-1}x^{2n-2}}$$ ??
1
vote
1answer
88 views

Inequality for line integral

Let $F(x)$ be a continuous (not necessarily monotonic) function defined on smooth curve $C$. I am wondering if the following inequality holds for line integrals. $$|F(a)-F(b)|\leq \int_C ...
1
vote
0answers
51 views

Measurable injection from $C[0,1]$ to $[0,1]$

Apparently such a map exists, where $C[0,1]$ is equipped with the Borel $\sigma$-algebra induced by the uniform norm and $[0,1]$ the usual Borel $\sigma$-algebra. This seems very surprising---that ...
4
votes
1answer
87 views

$\overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$

Let $X$ and $Y$ be two arbitrary subsets of $\mathbb{R}$. Show that $\overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$ Proof since $X\cap Y \subseteq X$ and $X\cap Y \subseteq Y$ ...
1
vote
2answers
172 views

Remember the Christoffel symbols

This might be a little bit different from what is asked normally on this page, but does anybody here know a good way to remember the definition of the Christoffel symbol? \[ \Gamma^k_{ij} = \frac 12 ...
2
votes
1answer
197 views

Natural progression in a curriculum for self-study of analysis

Would you list what is a natural and effective progression to self-study topics in analysis in order to gain a broad knowledge of the enormous corpus of knowledge that modern analysis involves. As a ...
0
votes
1answer
83 views

Use $\sum\limits_{n=-\infty}^{\infty}\frac{1}{n^2+a^2}$ to evaluate $\sum\limits_{n=0}^{\infty}\frac{1}{n^2+a^2}$

I want to evaluate the series $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n^2+a^2}$ with $a>0$. I know that $\displaystyle \sum_{n=-\infty}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{a}\coth \pi a$ ...