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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0
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1answer
14 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
0
votes
1answer
60 views

How to prove this limit of derivative to zero [on hold]

This is a test question in real analysis and I need help to prove it. Let $f:\mathbb{R}\to\mathbb{R} \in C^{\infty}$ be periodic of period $1$ and nonnegative. Show that $$ ...
3
votes
2answers
230 views

Dominated Convergence Theorem for Sets

This was an interesting question, which gives the analog of the better known Dominated Convergence Theorem for Lebesgue integrable functions. Suppose $E_{n}\to E$ pointwise (e.g. the indicator ...
1
vote
2answers
150 views

Any idea on this problem $\lim \limits_{x\to\infty}f(x)=0$ [on hold]

This is a question in real analysis. I think it needs ODE to prove, but not sure. Any idea is welcome. Let $f$ be a real valued continuous function on $[0,\infty]$ such that $$ \lim ...
0
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0answers
14 views

Help in finding a paper on nonlinear Schrodinger equations

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
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0answers
27 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
-1
votes
1answer
23 views

Antiholomorphic function

Let f be an antiholomorphic function in C. $z_0 \in C - C(0,1). $ Show that $\frac{1}{2 \pi i}\oint \frac {f(z)}{z-z_0} = \begin{cases}f(0) &\text{for } |z_0| < 1\\f(0) - f(\frac{1}{z_0}) ...
2
votes
1answer
36 views

Proving that a trigonometric sum is in $L^2$

how can I use Perseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? thank you!
0
votes
1answer
30 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
0
votes
1answer
49 views
+50

The closure of an open set in $\mathbb{R}^n$ is a manifold

I want to solve the following exercise from M. Spivak's Calculus on Manifolds (p. 114): (a) Let $A \subseteq \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n-1)$-dimensional manifold. ...
1
vote
0answers
22 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
2
votes
1answer
63 views

A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Theorem: Consider , $$F(x)=\int_a^xf(t)\,dt$$ If the function $f:[a,b]\to \mathbb R$ is continuous then $F(x)$ is differentiable and $F'(x)=f(x).$ I know that the continuity condition ...
2
votes
2answers
148 views

If an IVP does not enjoy uniqueness, then there are infinitely many solutions.

I am trying to prove than when an IVP has more than one solutions, then there exist infinitely many different solutions. I know that when the Lipschitz condition holds, there is at most one solution ...
0
votes
1answer
43 views

A necessary condition for a multi-complex-variable holomorphic function. [on hold]

Let $\Omega\subset \mathbb{C}^n$ be an open unit ball, $f:\Omega \to\mathbb{C}$ is a bounded function. For $a \in \mathbb{C}^n$, define $$ ...
0
votes
1answer
28 views

If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$.

TRUE or FALSE: If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$. My Proof: Since $f$ is convex function so, $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ , for all ...
0
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0answers
18 views

Prove that the function $ξ\in R \mapsto {e^{i\cdot ξ\cdot λ}-1\over i\cdot ξ}-λ$ is $C^{\infty}$ is $C^{\infty}$

Prove that the function $$ξ\in R \mapsto {e^{i\cdot ξ\cdot λ}-1\over i\cdot ξ}-λ$$ is $$C^{\infty}$$ (and in the point of ξ=o) Any ideas how to prove this? i am trying to think some ideas but i can ...
-3
votes
1answer
22 views

Show, directly from the definition, that the following series is convergent. [on hold]

Using the definition of a convergent series, how do you show that the series $\sum_{n=1}^{\infty} (\frac{-2}3)^n $ converges.
0
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0answers
31 views

Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
1
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0answers
12 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise if $f:[0,1]\to \mathbb R$ is only piecewise continuous (but ...
0
votes
2answers
23 views

Recurrence sequences with two initial condition: how do I calculate the limit?

I've done some exercises with recurrence sequences with one initial condition. So, now that I'm attempting one exercise with two initial conditions I'm confused. Could you show me what to do? Let ...
1
vote
1answer
39 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
0
votes
1answer
35 views

Bring a proof for the fundamental theorem of calculus.

If $f\in \mathscr{R}$ on $[a,b]$ and if there is a differentiable function $F$ on $[a,b]$ such that $F'=f$, then $$\int_a^b f(x)\ \ d(x)=F(b)-F(a)$$
1
vote
1answer
292 views

Analysis - Fourier Transforms - show that convolution of characteristic functions is continuous

I would appreciate any instruction on the following exercise from real and complex analysis: Suppose $A$ and $B$ are measurable subsets of $\Re^1$, having finite positive measure. Show that the ...
19
votes
1answer
495 views

Is there a function having a limit at every point while being nowhere continuous?

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
-2
votes
0answers
29 views

How can the elements $a_1, a_2, a_3\ldots, a_n$ be distinct in Theorem 2.13 of Rudin?

In Theorem 2.13 of Rudin, how could the elements $a_1,a_2,\ldots, a_n$ be distinct like he says they can? $A$ is a countable set (or just a set) and, therefore, all elements must be distinct. Perhaps ...
0
votes
1answer
13 views

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?
2
votes
1answer
69 views

Hausdorff Dimension of a manifold of dimension n?

Let's say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus ...
8
votes
3answers
5k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
5
votes
1answer
78 views

Prove $ \lim\limits_{n\to\infty}\int_0^1 f(x)g(nx)\,dx=\int_0^1 f(x)\,dx\int_0^1 g(x) \, dx $ [duplicate]

Let $f$ and $g$ be a real valued continuous functions on $\mathbb{R}$ such that $f(x+1)=f(x)$ and $g(x+1)=g(x)$ for all $x\in \mathbb{R}$. Prove that $$ \lim_{n\to\infty}\int_0^1 ...
-1
votes
3answers
52 views

Area of a region under the mapping $f$

Consider the function $f:\mathbb R^{2} \to \mathbb R^{2}$ given by $f(x,y)=\left(e^{x+y},e^{x-y}\right)$. Area of the image of the region $\{(x,y)\in \mathbb R^{2} | 0<x,y<1\}$ under the mapping ...
1
vote
1answer
51 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
0
votes
2answers
41 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
2
votes
1answer
45 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
3
votes
1answer
16 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
0
votes
1answer
12 views

Find limit inferior and limit superior of $[1+\sin n]$ and $n - [\sqrt n]$

I have to find the limit inferior and limit superior of the following sequences: $$[1+\sin n]$$ and $$n - [\sqrt n].$$ I have done similar exercises before, but never with the integer part function ...
2
votes
1answer
26 views

derivative of $f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi)$

Let $$ f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi) $$for some $r\in(0,\sigma)\subset\mathbb R$ and $\phi\in (0,\rho)\subset(0;2\pi]$. How do you calculate $Df=(\partial_1 f,\partial_2 f)$ ? I thought ...
0
votes
1answer
14 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
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votes
0answers
28 views

A problem of Taylor series [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
0
votes
2answers
24 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
0
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0answers
5 views

Bipartite graph matching partitioning using clustering algorithm

I am identifying information from a document using bipartite graph model now I have to extract that information which are closely matched. hence I want to use clustering technique to group the data ...
8
votes
1answer
237 views

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
0
votes
2answers
166 views

How can I investigate the differentiability of this function?

I leanred, if all partial derivatives exist and all are continuous, then it is differentiable. Am I wrong? I tried same way for this problem, I think it is differentiable because all the derivative ...
0
votes
3answers
99 views

What is the limit of $\mathrm{e}^{At}$ when $t\to \infty$, for $A$ a square matrix? [on hold]

I was doing a matrix calculation and need to find $$\lim_{t\to \infty} \mathrm{e}^{At}=?$$ What is the limit of $\mathrm{e}^{At}$ when $t\to \infty$, for $A$ a matrix?
10
votes
6answers
3k views

limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$

I am trying to solve the following problem at the level of a senior undergrad analysis level. So, the problem is as follows: We are given a function $f$ which is continuous on the interval $\left [ ...
4
votes
0answers
43 views

How to prove this integral [duplicate]

Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1) $$
1
vote
1answer
45 views

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$, such that: \begin{equation*} f(x)=0, \textrm{ if } x\in ...
4
votes
3answers
47 views

Find an example of a sequence not in $l^1$ satisfying certain boundedness conditions.

This question is about getting a concrete example for this question on bounded holomorphic functions posed by @user122916 (something that he really expected as explained in the comments). Give an ...
2
votes
1answer
30 views

Relation between runge domain and polynomial convexity

Are these concepts the same? Just to state the definitions Definition 1 A domain $\Omega \in \mathbb{C}^n$ is a Runge domain if every function $f \in H(\Omega)$ can be approximated, uniformly on ...
1
vote
1answer
26 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
6
votes
1answer
371 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...