Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
157
votes
13answers
11k views
36
votes
14answers
5k views
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$$
Well, can anyone ...
37
votes
3answers
2k views
Why is $1^{\infty}$ considered to be an indeterminate form
From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
9
votes
4answers
1k views
Sine function dense in $[-1,1]$
We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
34
votes
5answers
3k views
Why does $1+2+3+\dots = {-1\over 12}$?
$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$
$\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ ...
16
votes
7answers
3k views
Proof for an integral involving sinc function
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also ...
10
votes
5answers
1k views
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
17
votes
1answer
1k views
Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
Here is an exercise, on analysis which i am stuck.
How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on ...
12
votes
3answers
909 views
Upper and Lower Bounds of $\emptyset$
From some reading, I've noticed that $\sup(\emptyset)=\min(S)$, but $\inf(\emptyset)=\max(S)$, given that $\min(S)$ and $\max(S)$ exist, where $S$ is the universe in which one is working. Is there ...
12
votes
3answers
553 views
$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$
Trying to solve
$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.
I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$
Would ...
3
votes
5answers
298 views
$\lim_{n \to +\infty} n^{\frac{1}{n}} $
I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem?
$$
\lim_{n \to +\infty} ...
18
votes
2answers
732 views
A stronger version of discrete “Liouville's theorem”
If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition
$$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, ...
1
vote
3answers
231 views
Prove that a function having a derivative bounded by 0.49 has a unique solution $\frac{2x+\sin(x)}{2}$
Let f : $\mathbb{R} \to \mathbb{R}$ be differentiable function and suppose that $|f'(x)|\le 0.49$ for all $x \in \mathbb{R}$. Prove that the equation $f(x) =\frac{2x+\sin(x)}{2}$
has a unique ...
15
votes
4answers
1k views
Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$
While trying some problems along with my friends we had difficulty in this question.
True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
3
votes
3answers
402 views
Combinatorial proof
Using notion of derivative of functions from Taylor formula follow that
$$e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$
Is there any elementary combinatorial proof of this formula
here is my proof for ...
4
votes
2answers
639 views
How do you show that $l_p \subset l_q$ for $p \leq q$?
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
5
votes
1answer
452 views
Inequality involving $\limsup$ and $\liminf$
This may have been asked before, however I was unable to find any duplicate.
This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14:
If $(a_n)$ is a sequence in ...
15
votes
4answers
563 views
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?
I think it is probable because we can consider
$$ y ...
7
votes
4answers
572 views
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ...
12
votes
1answer
510 views
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
Maybe I would have to use the Rademachers.
5
votes
3answers
478 views
Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$
How does one sum the series, $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$
This was asked to me by a high school ...
6
votes
5answers
538 views
Defining division by zero
I have looked through some of the previous questions posted on this topic, and I think mine is different.
Is there a flaw in defining division by zero? For example, define
$\frac{a}{0} = \infty_a$ ...
5
votes
5answers
1k views
If $f'$ tends to a positive limit as $x$ approaches infinity, then $f$ approaches infinity
Some time ago, I ask this here. A restricted form of the second question could be this:
If $f$ is a function with first derivative continuous in $\mathbb{R}$ and such that $$\lim_{x\to \infty} ...
2
votes
2answers
140 views
prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$.
Suppose that $\lim_{x\rightarrow a}f'(x)$ exists.
Then, prove that $f$ is differentiable at $a$ and ...
1
vote
2answers
169 views
prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:
$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$
$Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$
$p=\int_{-1}^{1}Z(x)dx$
...
1
vote
2answers
346 views
$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology
From Wikipedia
if $\{A_n\}$ is a sequence of subsets of a topological space $X$,
then:
$\limsup A_n$, which is also called the outer limit, consists of those
elements which are limits of ...
17
votes
1answer
677 views
A smooth function's domain of being non-analytic
I am wondering how much a smooth function may be non-analytic, because in proofs, whilst there non-analytic smooth functions, it would suffice if a smooth function were analytic on only a "small set". ...
13
votes
3answers
707 views
Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.
Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$
Computing the Fourier series of $f$ and ...
14
votes
4answers
546 views
Evaluation of the integral $\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$
How can I evaluate the integral
$$\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$$
I tried manipulating the known integral
$$\int_0^1 \frac{\ln(1 - x)}{x}dx = -\frac{\pi^2}{6}$$
but couldn't do anything with ...
9
votes
1answer
209 views
Applications of Pseudodifferential Operators
I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
8
votes
3answers
564 views
Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small?
I started thinking about the number of associative (binary) operations on a set with $n$ elements today. Looking online I found this paper which indicates only $113$ of the possible $19,683$ ...
15
votes
2answers
555 views
Continuity and the Axiom of Choice
In my introductory Analysis course, we learned two definitions of continuity.
$(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
15
votes
1answer
522 views
If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?
If two analytical functions of $\mathbb{C}$ f and g are equal on an infinite number of input values, than they are equal. I can't seem to find a counterexample, but I haven't seen this anywhere ...
6
votes
1answer
171 views
Proof that a certain entire function is a polynomial
Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this ...
3
votes
2answers
370 views
Uniform continuity
I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is bounded.
Thank you.
11
votes
4answers
759 views
Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$
How do I show that:
$$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$
This is actually problem B $4371$ given at this link. Looks like ...
9
votes
2answers
400 views
Characterising Continuous functions
We know that if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function, then $f$ carries connected sets to connected sets and compact sets to compact sets. That is if $A \subset \mathbb{R}$ is ...
19
votes
2answers
555 views
Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility
I was told by a tutor that if $f: \mathbb{R}^n \longrightarrow \mathbb{R}^n$ has an invertible Jacobian Matrix for all $x \in \mathbb{R}^n$ and $\lim_{|x_k| \rightarrow \infty}|f(x_k)|=\infty$ for all ...
10
votes
5answers
467 views
Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$
If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
14
votes
2answers
417 views
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere.
If there exists $c \in \mathbb{R}$ ...
11
votes
2answers
339 views
Choice of $q$ in Baby Rudin's Example 1.1
First, my apologies if this has already been asked/answered. I wasn't able to find this question via search.
My question comes from Rudin's "Princicples of Mathematical Analysis," or "Baby Rudin," ...
8
votes
2answers
338 views
How does Lambert's W behave near ∞?
How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of
$$\frac{W(x)}{\ln(x)}$$
near $\infty$ (but along the positive real line, if that ...
6
votes
0answers
264 views
On the weak and strong convergence of an iterative sequence
I have some difficulties in the following problem.
I would like to thank for all kind help and construction.
Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
8
votes
4answers
379 views
Summation of $\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)}$
For $x>0$ and $y>x+1$, how do we prove that $$\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)} = \frac{x}{y-x-1}$$
7
votes
1answer
371 views
Transfinite series: Uncountable sums
If you sum an expression over an uncountable set
$\sum_{x\in \mathbb{R}}f(x)$, then do we need $f(x)=0$ on all but a countable subset in order for the sum to have a finite value?
If not can you give ...
4
votes
1answer
547 views
Is there an Inverse Gamma $\Gamma^{-1} (z) $ function?
Since $\Gamma$ is not one to one over the complex domain, Is it possible to define some principal values ( analogues to Principal Roots for the Root function ) so we can have a $\Gamma^{-1} (z)$ ...
2
votes
4answers
783 views
Why if $f'$ is unbounded, then $f$ isn't uniformly continuous?
I've $I = [0 ; +\infty) $ and $f: I \rightarrow \Bbb R.$
a. I've proved that if $f'$ is bounded on $I$ then $f$ is uniformly continuous on $I$.
b. I've proved that if $\lim f' = \infty$ ...
8
votes
3answers
534 views
How to prove $(1+1/x)^x$ is increasing when $x>0$?
Let $F(x)=(1+\frac{1}{x})^x$.
How do we prove $F(x)$ is increasing when $x>0$?
3
votes
4answers
188 views
Prove that the sequence $(n+2)/(3n^2 - 1)$ converges to the limit $0$
I need to prove that
$$\lim_{n \to \infty} \frac{n+2}{3n^2 - 1} = 0.$$
I usually have no trouble with these types of proofs but the $-1$ in the denominator and the $n+2$ are making things a little ...
2
votes
3answers
195 views
What is the expression for this summation?
Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
