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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0answers
49 views

when we can say that $\limsup_{n\to\infty}f(a_n)=f(\limsup_{n\to\infty}a_n)$?

when we can say that $\limsup_{n\to\infty}f(a_n)=f(\limsup_{n\to\infty}a_n)$? (What conditions must have the function $f$?)
1
vote
0answers
25 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
4
votes
1answer
204 views

Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
1
vote
0answers
42 views

Roots of this trigonometric polynomial

Let $f:[0,2\pi) \rightarrow \mathbb{R}$ with $f(x):=\sum_{n=0}^{k}a_n \left(1+\cos(x)\right)^n$ for arbitrary $a_n$ with $a_k \neq 0$. My question is: What is the maximum number of zeros that this ...
3
votes
1answer
27 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
73 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
2
votes
1answer
97 views

Sequence of orthogonal vectors in a Hilbert space

Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequance of pairwise orthogonal vectors in a Hilbert space $H$. Show that the following are equavalent: (a) $\sum_{n=0}^\infty x_n$ converges in the norm topology ...
4
votes
1answer
207 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit $...
3
votes
2answers
225 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
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vote
1answer
65 views

Does the following sequence converge?

Suppose $a_i>0$ for all $i$, $\frac{\sum_{i=1}^n a_i}{n}\to \infty$ and p>1. Let $$y_n = \frac{(\sum_{i=1}^n a_i)^p}{n^{p-1}\sum_{i=1}^n(a_i^p)}.$$ Is $y_n$ monotonic? How can you prove or disprove ...
4
votes
2answers
157 views

Solving a 2nd order nonlinear ODE

Could you help me solve or give me some advice about following differential equation $$ 2(y')^2 + 3xy'y'' + 3yy'' = 0 $$
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vote
2answers
288 views

Difference between the two definitions about the equality of two functions

From a long time I have found there are two definitions about the equality of two functions (or identity of two functions). I quoted the two definitions in the following: Zorich's definition (Zorich,...
1
vote
1answer
95 views

Is there a differentiable function f which the differential function f' is bounded but has no maximum on a closed interval.

Is there a differentiable function $f$ in which the differential function $f'$ is bounded but has no maximum on one closed interval? Thanks
1
vote
2answers
192 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times \...
1
vote
1answer
44 views

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$ Prove that if $f(\mathbb{D})⊂\mathbb{D}$ or $\mathbb{D}⊂...
0
votes
2answers
72 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
3
votes
2answers
291 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) \...
2
votes
1answer
85 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
0
votes
2answers
114 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
2answers
54 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) =...
1
vote
1answer
339 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
1
vote
1answer
33 views

Relationship between big O notation and exponential type

Let $f: \mathbb{R} \to \mathbb{R}$, $C\in \mathbb{R}$. What, if any, is the difference between "$ f = O(e^{Cx}) $" and "$f$ is of exponential type $C$"? If they're different, is it possible to ...
1
vote
1answer
100 views

How to interchange limit and integral?

Suppose $f_{n}, f\in L^{1}(\mathbb R)$ with the properties that, $f_{n}(x)\to f(x)$ point wise for each $x\in \mathbb R;$ $\|f_{n}\|_{L^{1}(\mathbb R)} \leq \|f\|_{L^{1}(\mathbb R)}$ for every $n\...
0
votes
1answer
112 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
1
vote
1answer
99 views

Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
4
votes
2answers
282 views

An “obvious” statement about a nonincreasing supremum

Consider a nonnegative function $f(t,x): [0,\infty) \times [0,1] \rightarrow [0, \infty)$. Suppose we have the following property: $$ \mbox{ If } ~~~~~~~~~~f(t,y) > \frac{1}{2} \sup_{x \in [0,1]} ...
2
votes
0answers
74 views

Show a set is dense in $C(X)$

Let $X$ be a totally discontinuous compact space. Show that the algebra generated by $$\{f_F; ~f_F=\chi_F-\chi_{X/F},F \text{ is a clopen subset of }X\}$$ is dense in $C(X)$. My attempt: Suppose $\{...
4
votes
1answer
103 views

Is this another version of the gamma function?

I know that $\Gamma \left( x \right) $ is the unique function on $x \in (0, \infty)$ such that $f \left( 1 \right) =1$ $f(x+1)=xf(x)$ ${\frac {d^{2}}{d{x}^{2}}}ln(f \left( x \right))>0$ ...
1
vote
0answers
34 views

Show that $ \prod_{k=1}^n [1 + p_{nk}(e^{it} - 1)] \rightarrow e^{\lambda(e^{it} - 1)}, n \rightarrow \infty $

Suppose that $0\leq p_{nk} \leq 1, 1 \leq k \leq n$, $\max_{1 \leq k \leq n} p_{nk} \rightarrow 0, n \rightarrow \infty$ and $\sum_{k=1}^n p_{nk} \rightarrow \lambda$. Show that $$ \prod_{k=1}^n [1 ...
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vote
0answers
43 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
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vote
1answer
70 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
4
votes
2answers
111 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
0
votes
1answer
41 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < \...
0
votes
2answers
54 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
0
votes
1answer
116 views

How to interchange sum and integral?

We fix the point $\xi_{0}\in \mathbb R.$ Choose sequence $\{f_{n}\}_{n\in \mathbb N}\subset L^{1}(\mathbb R)$ with the following property : (1) $\|f_{n}\|_{L^{1}(\mathbb R)} \leq 1, $ for $n\in \...
0
votes
1answer
55 views

Prove two solutions of differential equation are the same

In a recent work I had to solve the following differential equation: $$ r x''(r)+r x'(r)^2+x'(r)-\frac{4}{r}=0~~. $$ To do so I used two methods and I got, using each, two solutions with different ...
3
votes
3answers
690 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
1
vote
1answer
69 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
1
vote
1answer
68 views

How to find an everywhere discontinuous real function with $F((a+b)/2)<(F(a)+F(b))/2$?

In here I posted a non-constructive everywhere discontinuous real function with $$F((a+b)/2)=(F(a)+F(b))/2$$ based on the using of Hamel basis. And Conifold answered there that there is no explicit ...
2
votes
4answers
456 views

Analysis question limit sin function as n goes to infinity [closed]

can you help me with the following: $\lim_{n \rightarrow \infty} \sin^{2} \pi \sqrt{n^2 + n}$ Thanks a lot!
1
vote
1answer
30 views

Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
2
votes
1answer
130 views

Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
2
votes
0answers
55 views

Discontinuous linear operator on $\ell^{2}$

Let $e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots)$ where $1$ is in the $n$th position. Then $\{e_{n}\}$ is an orthonormal basis for the Hilbert space $\ell^{2}(\mathbb{N})$. Does there exists a linear ...
2
votes
2answers
158 views

Asymptotic Behaviour Of A Bizarre Function

It is relatively easy to show that the asymptotic behaviour of $f(x)$, where $$ f(x)= \left[\frac{x}{2}\right] + \left[\frac{x}{4}\right] + \left[\frac{x}{8}\right] + \left[\frac{x}{16}\right] + \...
1
vote
0answers
124 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
3
votes
1answer
105 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to \...
2
votes
1answer
104 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb R^...
1
vote
0answers
333 views

Absolutely continuous function admits weak derivative

How to prove that an Absolutely continuous function admits weak derivative? Absolutely continuous function: Let $(X, d)$ be a metric space and let I be an interval in the real line R. A function $f: ...
4
votes
2answers
102 views

Riemann integral enigma

I tried to solve this problem from Souza Silva - Berkeley Problems In Mathematics: In the Solutions part, I founded next solution for this problem: I do not understand the last statement, so why $|...
4
votes
1answer
189 views

An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?

I would like to know an explicit method on constructing an everywhere discontinuous real function $F$ with the property: $$F((a+b)/2)\leq(F(a)+F(b))/2.$$ There is a non-constructive example (with the ...