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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102 views

Proof of $\displaystyle\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$

I want to prove $$\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$$ without useing L'Hôpital's rule.
9
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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
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0answers
111 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
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1answer
38 views

Test for uniform convergence on $\sum_{k=0}^\infty(\frac{1}{kx+2}-\frac{1}{(k+1)x+2})$

Test for uniform convergence on $0 \le x \le 1$: $$ \sum_{k = 0}^{\infty} \left[{1 \over kx + 2} - {1 \over \left(k + 1\right)x + 2}\right] $$ I think I'm supposed to use the weierstrass m-test. The ...
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1answer
20 views

Continuity along different spaces

1) Say I have a function that is continuous along $\mathbb{R}.$ Would that function be then continuous along $\mathbb{Q}$ ? How about the other way around? 2) If I have two functions that are not ...
2
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1answer
195 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
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3answers
151 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
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293 views

To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.

Let $f$ be a compactly supported nonnegative $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2\leq C f(x) $ by showing that for every point $...
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1answer
56 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
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0answers
34 views

ODE Initial value problem formualtion

If I have the following ODE initial value problem, $$\begin{align} y'(t) &= f(t), \quad t>0, \\ y(0) &= y_0. \end{align}$$ Then I was taught that a solution to the problem is given by ...
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1answer
38 views

Showing $f_{n} \rightarrow f$ in $L^{1}$ given an integral condition

Let $f_{n}: [0, 1] \rightarrow [0, \infty)$ be a Borel measurable function such that $$\int_{0}^{1}f_{n}(x)\log(2 + f_{n}(x))\, dx < \infty.$$ If $f_{n} \rightarrow f$ Lebesgue almost everywhere. ...
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0answers
14 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
3
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2answers
349 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family $\...
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2answers
298 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
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1answer
47 views

Integral inequality with gamma function

I have some trouble with paper I'm reading. The goal is this: let $s=\frac{1}{2}+\frac{1}{\log n}+it$. $M$ is a function such that $M(s)=O(\log^{3}(N(|t|+2)))$. Define $$U(s)=\frac{1}{2\pi i}\int^{\...
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1answer
114 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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1answer
217 views

Upper semicontinuous function and equivalent statements

Problem Let $f:\mathbb R^n \to \overline{\mathbb R}$, then the following statements are equivalent: (1) $f$ is upper semicontinuous; (2) for every $t \in \overline{\mathbb R}$, $\{x \in \mathbb R^n:...
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0answers
143 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
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1answer
126 views

Existence of solution of ordinary differential equation

I am reading a proof of the existence of solutions for ordinary differential equations and I have some basic doubt. I'll copy the statement, the part of the proof I don't understand and my question: ...
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2answers
97 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
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3answers
545 views

How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?

Question: Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$ I think this ...
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2answers
569 views

Multivariable version of the extreme value theorem

The Wikipedia entry on the extreme value theorem says that if $f$ is a real-valued continuous function on a closed and bounded interval $[a,b]$, then $f$ must attain a maximum value, i.e. there exists ...
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1answer
52 views

Understanding an equality that involves the gamma function

$$\frac{1}{\sqrt{2\pi}}\sigma^m \Gamma\left(\frac{m+1}{2}\right) 2^{\frac{m+1}{2}} = 2^{\frac{m}{2}}\sigma^m \left(\frac{m-1}{2}\right) \left(\frac{m-3}{2}\right)\cdots \left(\frac{3}{2}\right) \left(\...
6
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1answer
179 views

Show that there is sequence of homeomorphism polynomials on [0,1] that converge uniformly to homeomorphism

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...
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1answer
50 views

Could not find equation of function from data

I have a problem to find equation of function of my data: 0.00 0.007 0.20 0.041 0.40 0.165 0.60 0.449 0.80 0.816 1.00 0.982 1.20 0.741 1.40 0.212 1.60 -0.362 1.80 ...
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 $f$?)
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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
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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 ...
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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
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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 ...
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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
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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
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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
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2answers
213 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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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 ...
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2answers
156 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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2answers
282 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,...
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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
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2answers
186 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
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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}⊂...
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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
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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 ...
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2answers
113 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 ...
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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) =...
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1answer
328 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 $\lambda$ ...
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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
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1answer
99 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
111 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
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1answer
98 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 ...