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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40 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
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1answer
86 views

How to show the convergence of this infinite series: $\frac{x}{1+x}- \frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}\dots$

My series is $$ \frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-..... $$ Given: $0<x<1$ I see that my nth term is $(-1)^{n+1} (\frac{x^n}{1+x^n})$ My approach was to use Dirichlet's test. ...
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1answer
36 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
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1answer
65 views

Prove that the series is convergent: 1- (1+1/3)/2 + (1+1/3+1/5)/3-…

I can see that this is an alternating series with the $n$-th term $$(-1)^{n+1}\frac{1+\frac13+\frac15+\cdots+ \frac{1}{2n-1}}{n}.$$ What test can I apply to show that it converges? Also, it ...
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1answer
42 views

Need my data to fit ANOVA…

I am doing some research on constructed wetlands. I have four wetland as follows. Wetland A - Gravels Wetland B - Gravels + Plant Wetland C - Biochar + Plant Wetland D - Biochar + Gravels + Plants ...
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1answer
140 views

Proving differentiability using Caratheodory's Lemma

Let $I$ be an open interval and let $c\in I$. Let $f:I\rightarrow\mathbb{R}$ be continuous and define $g:I\rightarrow\mathbb{R}$ by $g(x)=\left|f(x)\right|$. Prove that if $g$ is ...
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0answers
36 views

rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
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2answers
88 views

Compensation Question

I want to create a compensation system which takes into account two variables. Lets say I have $1M to distribute among ten employees who produce widgets. I want to compensate each employee by two ...
3
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1answer
90 views

Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
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1answer
68 views

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$.

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$. $(a)$ Show that $f(V) \subset V.$ $(b)$ Let $f_n$ be ...
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1answer
262 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
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1answer
37 views

Limit involving sinus to show resonance-behavior

I got the following term: $$ - \frac{\omega}{\mu^2 - \omega^2} \frac{1}{\mu} \sin(\mu t) + \frac{1}{\mu^2 - \omega^2} \sin(\omega t),$$ with $t, \mu \in \mathbb{R}$ and $\mu > 0$ and i'm ...
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1answer
49 views

Prove that $f(A\cap {B})\subseteq {f(A)\cap {f(B)}}$ [closed]

Let $f:S\to{T}$ be a function. If $A$ and $B$ are two arbitary subsets of $S$ prove that $f(A\cap {B})\subseteq {f(A)\cap{f(B)}}$
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1answer
87 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
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1answer
87 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
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1answer
37 views

common strategy for proving a real-valued function that is bounded

Yes I'm actually doing a prove on $$f(x)=\frac{x}{1+x^2}$$ but I'm not happy by only solving a particular case. So, i'm trying to do some conclusion here: I think one of the condition is to prove ...
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1answer
53 views

Convex combinations sequence question

Let $a_0,a_1,\beta$ be given with $0<\beta<1$ Let the sequence be defined by $a_{n+2} = \beta a_{n+1} + (1-\beta)a_n$ for $n\geq0$ Show that $\{a_n\}$ converges and find its limit.. How to ...
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1answer
110 views

Does existence of global minimum imply coercivity?

It is known that a coercive function over a closed, unbounded set has a global minimum. Is the converse true ? The larger context for this question is the following question: Suppose we are given a ...
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1answer
178 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
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0answers
38 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
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1answer
194 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
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1answer
116 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
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1answer
67 views

What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
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2answers
74 views

Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$
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1answer
56 views

Analysis Arithmetic series .Verify which of the following sequences converge.

Verify which of the following sequences converge.$$1.A(n)=\sum_{n=1,n=+00}(1/(n^{1+1/n})$$ $$2. B(n)=(1/\sqrt{n^2+1})+.......n/\sqrt{n^2+n}$$ $$3.C(n)=(n+cos(n^2))/(n+sin(n)) $$ .For the 3th one ...
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1answer
29 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
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198 views

Measure theory problem from Stein real analysis

Let $\mu$ be a Borel measure on the sphere $S^{d-1} = \{x \in \mathbb{R}^d:|x|=1\}$ which is rotation-invariant in the sense that $\mu(r(E)) = \mu(E),$ for every rotation $r$ of $\mathbb{R}^d$ and ...
2
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1answer
90 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
0
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1answer
111 views

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
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3answers
263 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
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1answer
137 views

Transforming ODEs into exact equations.

I want some examples of ODEs that can only be solved by transforming them into exact equations. They shouldn't be solvable with; Direct integration, separation of variables, manipulating a reverse ...
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2answers
147 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
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5answers
100 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
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1answer
229 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
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1answer
48 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
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1answer
66 views

How to arrive at desired equality?

Why is the following second equality true? $$e^{1+1/2+...+1/(n+1)} - e^{1+1/2+...+1/n} \\= ...
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0answers
50 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
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1answer
110 views

If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?

I was working my way through the set theory chapter in my Analysis textbook when I stumbled across these two theorems: Every infinite set has a countable subset A union of countable subsets ...
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1answer
78 views

Lemniscate curve parametrization exercise

Let $\gamma (t) : \mathbb R \to \mathbb R^2$ be the function $$\gamma(t)=\left(\frac{(1+t^2)t}{1+t^4},\frac{(1-t^2)t}{1+t^4}\right)$$ Prove that the function is $\gamma$ is differentiable, regular ...
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1answer
54 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
0
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1answer
61 views

Is there an explicit polynomial form for the product of consecutive integers?

I have the product $\prod_{j=0}^{r-1} (n+j)= n(n+1)\cdots(n+r-1)$ where n is a positive integer, and I was wondering if there was an explicit polynomial form for it (as a polynomial of degree r). I've ...
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0answers
72 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
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5answers
315 views

Assumptions in Word Problems (Calculus) [closed]

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
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1answer
160 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
0
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1answer
29 views

About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {$ {x∈X∣lim f_{n}∈R}$} $ $ = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞ ${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)}$ $ and that ...
1
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1answer
20 views

Is this a B-measurable funtion?

Is the function defined by: $f(x)=e^x $ if x is in E and $f(x)=e^{-x} $ if x is not in E measurable?, here f goes from R to R, and E is not member of the Borel-algebra
2
votes
2answers
156 views

Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
1
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2answers
46 views

Prove that $f(x) > g(x)$ where both functions are convex and have the same value and slope at $0$

Let $f: [-a,a] \to \mathbb{R}$ and $g: [-a,a] \to \mathbb{R}$ be two non-negative, convex and smooth functions. We further know $f(0) = g(0)=0$ and $f'(0) = g'(0)=0$. I'd like to show $$f(x) \ge ...
1
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2answers
72 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
1
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2answers
101 views

How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...