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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1answer
17 views

Convex hull of $3$ dimensional set reduced to $2$ dimensional set

Let $S = \{(f_1(t), f_2(t), f_3(t)) : t \in \mathbb{R}\}$ and suppose $f_3(t) \geq 1$ for all $t \in \mathbb{R}$. Is finding the convex hull of $S$ in some way equivalent to find the convex hull of $T ...
3
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0answers
35 views

Let $h(z) = g(f(z))$. If two of the three functions $f$, $g$, and $h$ are holomorphic and non-constant, must the third also be holomorphic?

If $h$ and $g$ are holomorphic it seems like the answer is no. Let $f(z) = f(re^{i\theta}) = \sqrt re^{i\theta/2}$ for $\theta \in [0,2\pi)$, and let $g(z)=z^2$. Then $f$ is discontinuous on the ...
2
votes
4answers
88 views

Convergence of improper integral $\int_{0}^{\frac{\pi}{6}}\dfrac{x}{\sqrt{1-2\sin x}}dx$

I'm trying to determine whether the following improper integral is convergent or divergent. $$ \int_{0}^{\pi/6}\frac{x}{\sqrt{1-2\sin x}}dx $$ At first, I substituted $t=\dfrac{\pi}{2} - x $ and ...
0
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1answer
46 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a ...
0
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0answers
17 views

Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ ...
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1answer
49 views

Let $(s_n)$ be a sequence in $\mathbb{R}$. Prove $\lim_{n \to \infty}s_n=0$ if and only if $\lim_{n \to \infty}|s_n|=0$

First, assume that $\lim s_n=0.$ This implies that for any given $\epsilon > 0$, $\exists$ an $N$ such that for $n>N,|s_n-0|< \epsilon$. $|s_n-0|=|s_n|<\epsilon$ and $|s_n|=||s_n|-0| ...
1
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1answer
74 views

Help with Spivak's Calculus: Chapter 1 Problem 22

I've tried a lot of things, but I can't seem to get anywhere with this problem. I'm hoping the solution is simple but that I'm just missing it. The problem is as follows: Prove that if $y_0 \neq ...
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1answer
115 views

$ \int_{-\infty}^{+\infty}e^{-a^2t^2-b^2/t^{2}}\mathrm{d}t=\frac{\sqrt{\pi}}a\:e^{-2ab}$ [closed]

I want to show that $$ \int_{-\infty}^{+\infty}e^{-a^2t^2-b^2/t^{2}}\mathrm{d}t=\frac{\sqrt{\pi}}a\:e^{-2ab}. $$
3
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1answer
59 views

Find the derivative of the following by definition: $f(x,y)=(x^3, xy^2-y^2)$

$$f(x,y)=(x^3, xy^2-y^2)$$ So with these types of functions the derivative is $f'(x,y)=\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ ...
0
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1answer
24 views

If $X$ admits a nearest point to each point in every metric super space of $X$: every point like function on $X$ attains it's minimum value on $X$.

If $X$ admits a nearest point to each point in every metric super space of $X$, then every point like function on $X$ attains it's minimum value on $X$. Definitions used: Suppose $(X, d$) is a ...
0
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1answer
27 views

Constructing the Koenigs function about a repelling fixed point

My question is rather simple and I hope someone has some sort of an answer. I am looking for a simple yes or no answer, and a reference if anyone has one. We have a holomorphic function $f$ defined ...
3
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1answer
46 views

How would I find the second derivative of the bilinear $B(x,y)=Ax \times y$ where $A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$

$$B(x,y)=Ax \times y \text{ where } A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$$ Second derivative is obviously the first derivative of the first ...
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0answers
40 views

Standard properties of trigonemetric functions

You have $\sin(\frac{\pi}{6})= \frac{1}{2}$ and $\sin(\frac{5\pi}{6})= \frac{1}{2}$ and the interval [$\frac{\pi}{6},\frac{5\pi}{6}$] has length $\geq 1$ This is used as I'm sure most will be familiar ...
2
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1answer
58 views

Proofs that Dirichlet's function is not differentiable

Define $f: (0,1) \to (0,1)$ by $f(x)= \begin{cases} \frac{1}{q}, & \text{if $x=\frac{p}{q}$ in lowest terms with $p,q \in \mathbb{N}$} \\ 0, & \text{if $x$ is irrational} \end{cases} $ The ...
3
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0answers
25 views

Derivatives of (vanishing) infinite integrals

Probably this is a very stupid question, but if an infinite integral of some exponented expression vanishes, e.g.: $$\int_{\mathbb{R}^D} d^D \mathbf{x} \, P^r(\mathbf{x}) \cdots = 0,$$ does this imply ...
1
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1answer
28 views

Geometric meaning of the arc length function?

Let $[a,b] \subset \mathbb{R}$ and let $\varphi: [a,b] \to \mathbb{R}^n$ be continuously differentiable. Then the indefinite integral $x \mapsto \int_a^x \| \varphi'(t)\| \, dt$ on $[a,b]$ is the arc ...
1
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1answer
29 views

One-sided derivative of composition function

$f : V \subset\mathbb R^n \to \mathbb R$ is differentiable, $g : [0,1] \to V$ a continuous function. Given $g(1)=p, Df(p)=0$, and that $f\circ g $ is left differentiable, can we deduce that the left ...
0
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0answers
23 views

Linear decay of support of probability measure.

If $\int_{0}^{\infty}|x|dP(x)<\infty$, is it true that $\exists N,c$ s.t. $\forall n\geq N$, $$\int_{(cn,\infty)}^{\infty}|x|dP(x)<\frac{1}{n}?$$ No, for example ...
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5answers
57 views

Use definition of a limit to prove

Given $3$ functions $f,g,h:A \to \mathbb{R}$ and a cluster point $x_0$ for A such that $f(x)\leq g(x) \leq h(x)$, assume that $$\lim_{x \to x_0} f(x) = \lim_{x \to x_0} h(x)=L$$ By using ONLY THE ...
2
votes
2answers
57 views

Difference in definition of differentiation

Okay so quite often I see two different definitions of differentiation and I want to know when it is appropriate to use each one. $$\lim_{h \to 0} \frac{f(x_{0}+h)-f(x_{0})}{h}$$ and $$\lim_{x \to ...
4
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0answers
63 views

Mathematical Olympiad Problem

Let $\Bbb{R}$ be the set of real numbers. Determine all functions $f:\Bbb{R}\longrightarrow \Bbb{R}$ satisfying the equation $$f(x+f(x+y))+f(xy) = x + f(x+y)+yf(x)$$ for all real numbers $x$ and $y$.
1
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2answers
32 views

bound of integrable function

I want to prove the following conjecture: if an integrable function $f(x)$ is continuous on (0,T] and unbounded at $x=0$, then there exists positive $M$ and $\alpha\in(0,1]$ such that $$ |f(x)|\leq ...
1
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1answer
16 views

Bounded second derivative bounds first at 1/2?

Let $f$ be a continuous real-valued function on $[0,1]$, which is twice-differentiable and satisfies $f(0)=f(1)$. Suppose that $M>0$ is such that $|f''(x)|\leq M$ for all $0<x<1$. Prove ...
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0answers
32 views

Prove that all the negative term of Laurent series is zero

If the Laurent series has isolated singularity $z_0 = 0$. Prove that all the negative term of Laurent series is 0. Can someone show me how to prove this problem. Thank you.
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0answers
25 views

Symmetry of the second derivative

For the purposes of this question, a function $f$ is differentiable at $x\in \mathbb{R}^d$ iff (i) the directional derivative $\mathrm{D}_vf(x)$ exists for all $v\in \mathrm{T}_x(\mathbb{R}^d)$ and ...
1
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1answer
31 views

Proving that $O(n)$ is compact

Let $O(n)$ denote the group of orthogonal matrices under multiplication. We want to show that this is set is compact. To show $O(n)$ is compact, we can use Heine-Borel and show that it is closed and ...
2
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2answers
68 views

Finding a particular type of sequence of functions

For every bounded function $f:[a,b] \to \mathbb R$ on a closed bounded interval $[a,b]$ , which is dis-continuous at at most countably many points of its domain ; can we find a sequence of ...
2
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1answer
37 views

Using basic definition to derive duplication formula for Gamma function

Baby Rudin chap 8, 8.21, some consequences of the gamma function, one of them is $$\Gamma(x)=\frac{2^{x-1}}{\sqrt\pi}\Gamma(\frac{x}2)\Gamma(\frac{x+1}2)$$ Rudin noted that this identity "followed ...
3
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1answer
57 views

Infinite series of integrals of $L^2$ functions

I'm hoping someone can help me with this integration problem I've been struggling with. Let $\{f_n\}$ be a sequence in $L^2(\mathbb{R})$ such that $\sum_{n=1}^\infty \lVert f_n\rVert^2_2<\infty$ ...
0
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1answer
35 views

How to find minimum value of this implicit function?

It is difficult for me to calculate this, what is the minimum $a$ such that $2|~x-y~|\leq a|~2x-3y^2-3~|$ holds for all $x,y\in R$ with $|~x~|<y$. equivalent to what maximum of $\frac ...
2
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1answer
24 views

select a nested sequence from a $G_{\delta}$ set

Here, page $457$, in the proof of theorem $1$, there is this sentence 'Select a nested sequence $\{ U(n,i,j)| j=1,2,... \}$, of open subsets of $I$ whose intersection is $f^{-1}[\frac{i}{n}, ...
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2answers
101 views

Riemann sum on infinite interval

It is well known that in the case of a finite interval $[0,1]$ with a partition of equal size $1/n$, we have: $$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} ...
0
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1answer
22 views

test the difference between two samples AND also control for other variables

I would like to test whether there are any difference in performance between two groups of samples $A$ ($100$ observation) and $B$ ($140$ observation), but at the same time, I would want to control ...
0
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0answers
54 views

Show that there is exactly one $z$ [duplicate]

Show that there is exactly one $z$ in the right-half-plane such that: $$z + e^{-z} = 2$$ I know somehow we have to use Rouche's theorem to show that there is exactly one root in the right half plane. ...
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3answers
60 views

Radon-Nikodym derivative of sum of two measures

Problem Statement: Suppose that $\mu$ and $\nu$ are two finite measures such that $\nu \ll \mu$, let $\rho = \mu + \nu$, and note that since $\mu(A) \le \rho(A)$, and $\nu(A) \le \rho(A)$, we have ...
0
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1answer
93 views

How to rigorously prove this limit

I would like to prove that: $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k\geq 0} f\left(\frac{k}{n}\right)\left(\left(1-\frac{\lambda}{n}\right)^k-e^{-\lambda\frac{k}{n}}\right)=0$$ where $f$ is a ...
0
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1answer
98 views

What does “span” looks like in infinite dimensional spaces? [closed]

I noticed that my prof loves to write $S = span\{v\}$ Instead of $\sum \alpha v$ or $a_1v_1 + a_2v_2+...$. Is he using "span" in a general way? What would span look like in infinite dimensional ...
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3answers
71 views

What is the difference between the closure of a set and a closed set?

Is "closure" and "closed" interchangeable? Can a set be a closure of a set without being closed?
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1answer
51 views

Continuity & topology question [closed]

If $f(x)= \sin x$ if $x$ rational and $0$ otherwise at what points is f continuous? Is it at all points that are integer multiples of $\pi$? Also on a side note if, working in R^2, a set A is ...
2
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2answers
59 views

The Fundamental Theorem of Calculus (for the lebesgue integral)

I am interested in the following statement. Let $f:[a,b]\rightarrow \mathbb{R}$ be continuous and differentiable. Then, if $f':(a,b)\rightarrow \mathbb{R}$ is integrable, then $$ f(x)-f(a)=\int ...
1
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1answer
31 views

Norm triangle inequality for convolutions proof

I'm trying to prove that $$\|f*g\|_{L_1}\le{\|f\|_{L_1}\|g\|_{L_1}}$$ with respect to a Haar measure over a group G. Using Fubini's theorem, I'm up to ...
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0answers
44 views

Particular map from a square to a parallelogram

I would like to present you a problem I have to solve. I don't think its solution is elementary, so any hint you can give me is really welcomed. Let's consider $Q_1$ the square in $\mathbb{R}^2$ of ...
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2answers
30 views

Why is this not a metric on the space of Riemann Integrable functions on [a,b]?

I've just started to teach myself a little on the basics of metric spaces, and came across the following question. Let $d_2$ be the pseudo-metric defined on the space of continuous functions on ...
5
votes
2answers
56 views

evaluate the integral

Evaluate the integral from: $$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx$$ The way I approach this problem is $$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx = ...
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0answers
7 views

Is it possible to distribute the result of dependent events amongst the events individually?

My example would come from blackjack, and my thinking is this: Hand = series of dependent events Value = Win/Loss amount in $ Hand = Value Player has a starting hand of 12, Dealer is showing a 7, ...
0
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1answer
34 views

Prove $\displaystyle \int_a^\infty f(x) \, dx \iff$ there exists $M_{\varepsilon}$

Show that $\displaystyle \int_a^\infty f(x) \, dx$ exists if, and only if, for every $\varepsilon>0$ there exists $M_\varepsilon$ such that $$\left|\int_s^t f(x) \, dx\right|<\varepsilon$$ for ...
1
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2answers
75 views

Evaluate the integral $PV\int_{-\infty}^{\infty} \frac{1}{(x^2+1)(x^2+2x+2)}dx$

Using the Cauchy Integral Principal to evaluate: $$PV\int_{-\infty}^{\infty} \frac{1}{(x^2+1)(x^2+2x+2)}dx$$ I know this integral has a pole at $x=i$, $x = -1-i$ and $x = -1+i$. Can someone please ...
3
votes
5answers
240 views

Algebra and Analysis [closed]

I just finished my first year of university, double majoring compsci and mathematics. My tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble ...
0
votes
0answers
15 views

Why $H_{V^* \cup W^*} > H_{V \cup W}$ if $H_V$ denotes entropy of language

Let $W \subseteq X^*$ be an infinite language over a finite alphabet $X$, and define ($|w|$ denotes the length of $w \in W$) $$ H_W := \limsup_{n\to \infty} \frac{\log_{|X|} | \{ w \in w \in W, |w| = ...
6
votes
1answer
96 views

calculate $\prod_1^\infty k^{\frac1{k!}}$

Is it possible to find the point of convergence of $\prod_1^\infty k^{\frac1{k!}}$ $K!=k(k-1)!$. My attempt: If $S_n=\prod_1^\infty k^{\frac1{k!}}$ then $\ln S_n=\sum_1^\infty \frac{\ln k}{k!}< ...