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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2
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1answer
62 views

Changing the values of a function $f:[a,b] \to \mathbb R$ of bounded variation for countably infinitely many points not a dense subset of $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a function of bounded variation. It is known that if we change its values at finitely many points of $[a,b]$, then the changed function still remains of bounded ...
3
votes
0answers
58 views

A integral inequality

Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\left( ...
1
vote
1answer
37 views

The behavior of $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ near $0$, where $\alpha \ge 1$.

Consider $\alpha \ge 1$. Let $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ and let $f(0)=0.$ In order to find the sign of $f'(x)$ when $\alpha \ge 1$ it is necessary to decide if ...
0
votes
2answers
108 views

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$ (I know the inverse of a closed set of a continuous function is closed, but is this a must?) And does the following apply to all ...
1
vote
1answer
52 views

Whether a real number is a dyadic rational iff its binary expansion terminates?

In self-studying a textbook on computability theory, I found that many of the exercises depend on the following factlet: A dyadic rational is a rational number whose denominator is a power of two, ...
4
votes
1answer
94 views

Find a formula for $f''$ in terms of $f$, where $f\gt 0$ and $(f')^2=f-\frac{1}{f^2}.$

Problem: Suppose that a function $f \gt 0$ has the property $$ (f')^2=f-\frac{1}{f^2} $$ Find a formula for $f''$ in terms of $f$. Hint: Use Theorem 7. Theorem 7: Suppose that $f$ is ...
-1
votes
2answers
45 views

Evaluation on a basis of gaussian integral

Knowing that $$\int_{-\infty}^\infty e^{-x^2} dx= \pi^{\frac{1}{2}}$$ Find: $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} dx$$ And my question is how does this help if have the ...
1
vote
1answer
26 views

How to show that this works (complex numbers)

So if I have a set of complex numbers: $A= \{z\in\mathbb{C} |\ \text{Re}\,(z) > 0, |z|<1\}$ So I have a problem showing this: For any $z\in A$ exists $w\in A$ such,that this works: ...
1
vote
1answer
38 views

Boundaries change in double integral

Calculate: $$\int_0^1 \int_0^{x^3} e^\frac{y}{x} dydx$$ Obviously i need to change it to $dxdy$ thus i need to change the boundaries of the second integral but how to do that in this case?
1
vote
2answers
23 views

Double integral on a compact subset

Calculate: $$\int \int _D \left(6x+2y^2 \right) dxdy$$ where D is a compact subset of $\mathbb{R}^2$ enclosed by a parabola $y=x^2$ and a line $x+y=2$. How to find that, how to find the limits of ...
1
vote
0answers
19 views

Generalized Hyperbolic and Circular Functions

I have recently posted a couple of questions in regards to Generalized Hyperbolic and Circular Functions and I was hoping to find a couple more papers available on the particular subject. The papers ...
2
votes
0answers
34 views

Rudin's Rank theorem

Rudin states the following: 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ is a $C^1$ mapping of an open set $E\subset R^n$ into $R^m$, and $F'(x)$ has rank $r$ for ...
2
votes
3answers
289 views

Double integral problem: $\int_0^\pi\int_x^\pi \frac{\sin y}{y} dy\, dx$

Calculate: $$\int_0^\pi \int_x^\pi \frac{\sin y}{y} dydx$$ How to calculate that? This x is terribly confusing for me. I do not know how to deal with it properly.
3
votes
2answers
58 views

find smallest $x>0$ such that $\frac{A}{cx}e^{-cx^2}\le \varepsilon$

I was estimating some error and I got $$\varepsilon(x)\le\frac{A}{cx}e^{-cx^2}$$ $A,c$ are known and positive, $x$ is also positive. The bigger the $x$ smaller the error. But I need to find the ...
2
votes
0answers
45 views

Inequality problem involving log function

Given $|f(x+y)-f(x)-f(y)| \leq x+y$ for all $x > y > 0$, prove that real valued function $f$ satisfies the inequality $|\frac{f(x)}{x} - \frac{f(y)}{y}| \leq M(1+\log_2\frac{x}{y})$ where M is ...
4
votes
0answers
62 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
0
votes
1answer
37 views

Bijective Conformal Mapping onto the Open Unit Disc $\mathbb{D}$

What is the explicit bijective conformal mapping $f(z):G_n\to\mathbb{D}$, $z\in\mathbb{C}$ for the following domain transformations: $G_1=\{x+iy~|~x>1/2,y>0\}$ is the open region of the first ...
1
vote
0answers
22 views

Find the $\epsilon - \delta$ values for the continuous function - modified step function defined on $[0,1]$

Let the modified step function be defined on $[0,1]$ by : $f(x) = \begin{cases} \bigg( \dfrac {2^n+1}{2}\bigg )x - \dfrac {2^n-1}{2^n} ; & n \in \mathbb N~~ , \dfrac {2} {2^n+1} ...
1
vote
2answers
38 views

Polynomial must be monotone between its extrema

Suppose that the polynomial function $f(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ has $k_1$ local maximum points and $k_2$ local minimum points. Show that $k_2=k_1+1$ if $n$ is even, and $k_2=k_1$ if $n$ is ...
3
votes
2answers
49 views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
0
votes
1answer
22 views

Equation of the plane tangent to the given surface

Find the equation of a plane tangent to the surface given by $$xyz+x^2-3y^2+z^3=14$$ at $$P=\left( 5,-2,3 \right)$$ In my opinion answer is: $$4x+27y+25z-41=0$$ If not please tell me what am i doing ...
2
votes
2answers
29 views

Use the lagrange's multipliers method to find a points on an ellipse

Question: Using the Lagrange's Multipliers method, find the points on the ellipse $x^2+2y^2=1$, that are situated in the longest and shortest distance from the line $x+y=2$. I know how to use ...
0
votes
0answers
16 views

proof coordinate functions of integrable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ integrable

If $$f(x)=f_1(x_1)\cdots f_n(x_n)$$ and $f$ is an integrable function from $\mathbb{R}^n$ to $\mathbb{R}$. Proof that $f_i(x_i)$, $i = 1, \ldots, n$ are integrable. With the Fubini theorem?
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0answers
20 views

Calculation of a Frechet derivative

Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real ...
1
vote
2answers
28 views

Function that fulfils the equation

Prove that for all $x\in\mathbb{R}$ there exists only one $y=y(x)$ that fulfils the equation: $$3x+e^x=y+e^y$$ I am completely lost with that. What should i do?
1
vote
0answers
16 views

Linear combination of the identity and function with nonnegative derivative

Given a arbitrary often differentiable functions $f\colon\mathbb{R}\rightarrow\mathbb{R}$, such that $f'(x)\ge 0$ for all $x$ holds. Define the function $$g\colon ...
0
votes
1answer
18 views

Problem with proving continuity of this function

As the title says I came across a problem with this function: $f:\mathbb{R}\to\mathbb{R}$ is continous $ f(x)=0 $, for $ x\in\mathbb{Q}$ So i want to prove that this works for every real number: ...
2
votes
2answers
38 views

boundness of first derivative of f

I think this problem is kind of a famous problem... A function f is real valued function from real line. Suppose that both of the absolute value of f and absolute value of the second derivative of f ...
1
vote
0answers
40 views

If $f$ is one to one show that $f(a) \in \partial \Omega$

Let $G$ be a region. Let $a \in G$. Suppose that $f:(G-{a}) \to \mathbb{C}$ is an analytic function such that $f(G-{a})=\Omega$ is bounded. i) Show that $f$ has a removable singularity at $z=a$ ii) ...
1
vote
1answer
24 views

Lower semicontinuous integer valued function

I remember reading in some book a characterization of lower semicontinuous functions that are integer valued (for example, rank of a matrix), along the lines that it can either not jump abruptly or ...
1
vote
2answers
71 views

Null set squared is a null set

I'm attempting to find a solution to the following problem that doesn't involve splitting this into various cases. The question is: "If $m^*(E) =0$, show that $m^*(E^2) = 0$, where $E^2 = \{x^2 ...
0
votes
0answers
31 views

Find all local maximum and minimum of $f$, which is $1$ if the decimal expansion contains a $5$ and $0$ otherwise.

Find all local maximum and minimum points of $f$. $f(x)= \begin{cases} 1, & \text{if the decimal expansion of $x$ contains a 5} \\ 0, & \text{otherwise} \end{cases} $ The answer states that ...
4
votes
1answer
45 views

A question on ordinary differential inequality

Could we find a solution $f=f(x)$ to the following initial problem for the OD inequality? $$3xf'+f-\sqrt{6f}\leq 0,\quad f(0)=0,\quad f(8/3)=6.$$ . Added: The above question is in fact a special ...
2
votes
4answers
71 views

Find the minimum value of $f(x)=\sum_{i=1}^n |x-a_i|$. [duplicate]

Let $a_1 \lt a_2 \lt \dots \lt a_n$. Find the minimum value of $f(x)=\sum_{i=1}^n |x-a_i|$. My guess is the minimum occurs at the middle point. However, I don't know how to show this since I can't ...
3
votes
0answers
36 views

gateaux derivative and frechet derivative

In calculus, we have the following equation $DF(x,y)=\partial F_xdx+\partial F_ydy$ if $F$ is differentiable. I think such equation still holds for frechet derivative, but not for gateaux derivative. ...
0
votes
1answer
40 views

$\Delta f=0$ in $\{x\in U:f(x)>0\}$ $\Rightarrow$ $\Delta f=0$ in $U$?

Let $f\geq0$ be a continuous function satisfying $\Delta f=0$ in $\{x\in U:f(x)>0\}$. I was wondering if one could follow $\Delta f=0$ in $U$, especially in the cases $f\in C^2$ or $\Delta f=0$ in ...
1
vote
1answer
26 views

Proving $f,g \in \mathbb D(U)$ (differentiable on $U$) $\implies f(x)g(x)$ is differentiable on U and $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$

This is done using compisition : $$x \mapsto^{F}(f(x),g(x))\mapsto^{B}f(x)g(x) \\ P(x)=B \circ F(x) \implies P'(x)h=B'(F(x))F'(x)h....(1)\\ \text{ I know that $B'(x)=B({}^1,\beta)+B(\alpha, ...
1
vote
0answers
12 views

when Wiener amalgam space is a subset of Lebesgue space?

Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= \|\hat{f}\|_{L^{p}}.$ In the definition of Wiener amalgam spaces $W(X, L^p)$, I am taking ...
1
vote
0answers
23 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

Let $\mathcal{L}$=space of all lines $L$. The X-ray transform is defined here: https://en.wikipedia.org/wiki/X-ray_transform $Xf:\mathcal{L}\to \mathbb{R}$ is defined by: $Xf(L)=\int_{t \in ...
0
votes
1answer
29 views

Analysis math irrational proof

How come $q = \frac{2p+2}{p+2}$ turns into $q^2 - 2 = \frac{2(p^2-2)}{(p+2)^2}$ I've tried factoring, square both sides, but I cant see it. this sample belongs to Rudin Analysis book
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vote
0answers
26 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
1
vote
2answers
101 views

About a limit-no l'hospital or derivatives

So i've come across this limit: $\lim_{x\to1} x^{\cot{\pi x}}$ So i found that its possible to solve without using those before-mentioned tools. I'm not sure how to do this limit,while i get stuck ...
0
votes
1answer
55 views

Point sets and limit points.

Show that if M is a point set having a limit point, then M contains at least 2 points. Must M contain 3 points? 4 points? Having difficulty describing and visualizing, because it seems rather ...
0
votes
0answers
26 views

The Divergence of The Elliptical Integral of First Kind $F(\phi,k)$

For what values of $k$ does the following elliptical integral of the first kind diverge? $$F(\phi,k)=\int\limits_0^{tan\phi} \frac{dt}{\sqrt{(1-t^2)(1-k'^2t^2)}}$$ where $\phi=\pi/4$ and ...
3
votes
2answers
70 views

inequality involving $x$, $x^3$,$\sin(x),\cos(x)$

Let $x \in \left[0,\dfrac {\pi} 2 \right]$. Prove the inequality $$6x \ge 6\sin x +x^3 \cdot \cos x$$ there is nice solution using Taylor expansion. Is there other one?
3
votes
4answers
125 views

limit involving $e$, ending up without $e$.

Compute the limit $$ \lim_{n \rightarrow \infty} \sqrt n \cdot \left[\left(1+\dfrac 1 {n+1}\right)^{n+1}-\left(1+\dfrac 1 {n}\right)^{n}\right]$$ we have a bit complicated solution using Mean value ...
1
vote
0answers
52 views

Is there an analytic approximation to integral of this form?

Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince $$\int_{0}^{t} ...
0
votes
2answers
25 views

Proving guess wrong used for substitution method

Following is my recurrence relation : $T(n) = 2T(n−1) + c_1$. Complexity: $O(2^N)$. I want to prove it by substitution method/ mathematical induction (You can get insight of it from : ...
0
votes
1answer
16 views

Analysis Proof- different conditions.

A continuous function on $[a,b]$ is also uniformly continuous on $[a,b]$. The following tries to illustrate what happens when the interval is not closed: Show: $f(x) = \frac{1}{x} $ is not ...
0
votes
1answer
31 views

Existence a diffeomorphism on $\mathbb R^n$

How may I show that for any $p,q\in\mathbb R$, there exist a diffeomorphism $F:\mathbb R\rightarrow\mathbb R$ such that $F(p)=q$ and $F$ to be an identity function outside of a some neighborhood of ...