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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5
votes
1answer
414 views

An Identity Concerning the Riemann Zeta Function

Let $\zeta$ be the Riemann- Zeta function. For any integer, $n \geq 2$, how to prove $$\zeta(2) \zeta(2n-2) + \zeta(4)\zeta(2n-4) + \cdots + \zeta(2n-2)\zeta(2) = \Bigl(n + ...
3
votes
2answers
206 views

Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$

Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (The factor of +1 in the ...
6
votes
3answers
679 views

Numerically estimate the limit of a function

Is there an algorithm that will allow me to numerically compute the limit of a function f(x) in a principled way? The most naive algorithm would be to continue to compute the function for larger ...
8
votes
4answers
413 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 $$\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)} = \frac{x}{y-x-1}$$
25
votes
8answers
5k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
13
votes
5answers
489 views

Applications of Fractional Calculus

I've seen recently for the first time in Special Functions (by G. Andrews, R. Askey and R. Roy) the definitions of fractional integral $$(I_{\alpha }f)(x)=\frac{1}{\Gamma (\alpha ...
7
votes
1answer
529 views

What is the Lebesgue mean of the fat Cantor set?

Everything that follows takes place in the Borel $\sigma$-algebra with Lebesgue measure. The Lebesgue mean of $f$ at $x$ is defined as $\displaystyle \lim_{\epsilon\to 0} ...
8
votes
3answers
2k views

Which metric spaces are totally bounded?

A subset $S$ of a metric space $X$ is totally bounded if for any $r>0$, $S$ can be covered by a finite number of $X$-balls of radius $r$. A metric space $X$ is totally bounded if it is a totally ...
14
votes
2answers
2k views

Limit of Nested Radical $\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots }}}$

How does one evaluate show that this limit: $$\lim_{n \to \infty}\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots \sqrt{1+(n-1) \sqrt{1+n}}}}}=3$$
4
votes
4answers
508 views

Partial sum of a given series

An Exercise from Apostol's Introduction to Analytic Number Theory which I am not able to solve. Let $\mathsf{S_{n}}$ denote the $n$-th partial sum of the series: $$\sum\limits_{r=1}^{\infty} ...
3
votes
4answers
431 views

Evaluating the improper integral $\int\limits_{0}^{\infty} \frac{x^{a-1} - x^{b-1}}{1-x} \ dx $

How does one evaluate the integral $$\int\limits_{0}^{\infty} \frac{x^{a-1} - x^{b-1}}{1-x} \ dx \quad \text{for} \ a,b \in (0,1)$$
3
votes
1answer
1k views

Numerically Solving a Second Order Nonlinear ODE

Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically. $$f''(R) + \frac{2}{R} f'(R)=\frac{0.7}{R} \left( \frac{1}{\sqrt{f(R)}} - \frac{0.3}{\sqrt{1-f(R)}} ...
4
votes
0answers
161 views

Critical exponents and point-wise convergence

A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. ...
12
votes
4answers
918 views

Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$

We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges. One can then ask the question of ...
0
votes
1answer
96 views

An inequality about norms of vector fields on Riemannian manifolds

Let $g_{ij}$ be the components of a symmetric rank-2 positive definite tensor (metric on a Riemannian manifold). Let ${C^i}$ and ${ \beta ^i }$ be components of a vector field on it, the former of ...
2
votes
1answer
150 views

Fitting object poses in 3D space

Using stereoscopic cameras, I track a certain object through it's path in space. For every frame, I compute it's pose in 3D. I can represent it's pose by either a translation vector + rotation matrix ...
4
votes
1answer
539 views

Number of Pythagorean Triples under a given Quantity

Consider the function $Pt(n)$. It tells us how many primitive Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an 'exact formula'; i.e. an ...
9
votes
1answer
1k views

Metric and Topological structures induced by a norm

While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm. 1) Is it possible to find two equivalent ...
16
votes
3answers
623 views

Different proofs of $\lim\limits_{n \rightarrow \infty} n \int_0^1 \frac{x^n - (1-x)^n}{2x-1} \mathrm dx= 2$

It can be shown that $$ n \int_0^1 \frac{x^n - (1-x)^n}{2x-1} \mathrm dx = \sum_{k=0}^{n-1} {n-1 \choose k}^{-1}$$ (For instance see my answer here.) It can also be shown that $$\lim_{n \to \infty} ...
1
vote
2answers
3k views

Example of a non measurable function!

Can we have a measurable function $f$, whose inverse is not measurable?
45
votes
5answers
5k views

Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
3
votes
1answer
128 views

The discrete Bessel kernel

Theorem 2 in a paper by Borodin, Okounkov, and Olshanski states that the discrete Bessel kernel $J(x,y,\theta)$ is given by $\sqrt{\theta} \frac{J_x J_{y+1} - J_{x+1} J_y}{x-y}$ where $J_x = ...
4
votes
2answers
873 views

Measurable function remaining constant

This is a problem which appeared in one of my tests, which i wasn't able to solve. Let $\Omega$ be a uncountable set. Let $S$ be the collection of subsets of $\Omega$ given by: $A \in S$ if and only ...
6
votes
4answers
860 views

Characterizing continuous functions based on the graph of the function

I had asked this question: http://math.stackexchange.com/questions/4412/characterising-continuous-functions some time back, and this question is more or less related to that question. Suppose we have ...
1
vote
3answers
794 views

Is a function analytical on C iff its Fourier-transform vanishes for negative frequencies?

I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication? edit for clarification: Is a function $f : \mathbb C ...
2
votes
1answer
308 views

Limit of the sequence $nx_{n}$ where $x_{n+1} = \log (1 +x_{n})$

Suppose $x_{1}>0$, and consider the sequence, $\{x_{n}\}$ defined as follows: $$x_{n+1}=\log(1+x_{n}) \quad n\geq 1 $$ Find the value of $\displaystyle \lim_{n \to \infty} nx_{n}$ I am having trouble ...
4
votes
4answers
716 views

What's the generalisation of the quotient rule for higher derivatives?

I know that the product rule is generalised by Leibniz's general rule and the chain rule by Faà di Bruno's formula, but what about the quotient rule? Is there a generalisation for it analogous to ...
71
votes
17answers
15k 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 ...
4
votes
3answers
349 views

Proving $\Bigl[\frac{8n+13}{25}\Bigr] - \Bigl[\frac{n-12 - \Bigl[\frac{n-17}{25}\Bigr]}{3}\Bigr]$ is independent of $n$

This an Exercise from Apostol's analytic number theory. I have been struggling with this problem for quite some time. Looks elementary though. The question is to prove that this quantity ...
6
votes
2answers
1k views

Countability of disjoint intervals

According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion: "Every collection of disjoint ...
3
votes
2answers
164 views

$f:X \rightarrow Y$. What are the conditions on $f$,$X$,$Y$ in order for an integral of $f$ to be defined?

Let's assume $X,Y$ are arbitrary sets, and $f$ is a function between them. What must those sets, and $f$, satisfy, in order for us to be able to define an integral of $f$ in a way that "makes sense"?
2
votes
2answers
358 views

Irrationality of $ \frac{1}{\pi} \arccos{\frac{1}{\sqrt{n}}}$

This paper arxiv.org/pdf/0911.1933 discusses, regarding the irrationality of certain trigonometric functions. Recently, i encountered this problem which says states the given function, $$ ...
12
votes
8answers
5k views

Why Circle encloses largest Area?

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
7
votes
2answers
709 views

Finding $\lim\limits_{n \to \infty} \sum\limits_{k=0}^n { n \choose k}^{-1}$

We know that $$ 2^n= (1+1)^n = \sum_{k=0}^n {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum_{k=0}^n {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$
16
votes
4answers
2k 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 ...
0
votes
1answer
517 views

Points of continuity of a function being dense

Suppose $X$ is a metric space. A real valued function $f$ defined on a metric space $X$ is said to be of first baire class, if $f$ is a pointwise, limit of a sequence of continuous functions on $X$. ...
0
votes
1answer
157 views

Finding $\alpha$ such that $f(\alpha(x+y))=f(x)+f(y)$

Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ ...
9
votes
2answers
547 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 ...
3
votes
1answer
74 views

For some sets $S\subseteq T\subseteq U$, when is $\inf_T S=\inf_U S$?

I've been struggling with the following problem I found for a while now: Suppose $(T,\preceq)$ is a partially ordered subset of $(U,\preceq)$ and $S\subseteq T$. If $\inf_T S$ and $u=\inf_U S$ both ...
2
votes
2answers
156 views

For $S\subseteq T\subseteq\mathbb{Q}$, it is possible for $\sup_T S$ to exist, but $\sup_\mathbb{Q} S$ does not?

I've been reading a bit about how the set of bounds changes for a set depending on what superset one works with. I considered the sets $S\subseteq T\subseteq\mathbb{Q}$ and worked out a few contrived ...
8
votes
3answers
639 views

Convergence of $\sum\limits_{n=1}^{\infty} \frac{1}{nf(n)}$

This problem is taken from Vojtěch Jarník International Mathematical Competition 2010, Category I, Problem 1. — edit by KennyTM On going through this post ...
3
votes
1answer
93 views

Does the existence of an infimum imply that the set of lower bounds of a set is totally ordered?

Say we have a partially ordered set $(S,\preceq)$, and some subset $E\subseteq S$ such that $E$ is bounded below and $\inf E$ exists. My question is, since $S$ is not totally ordered is it possible to ...
1
vote
2answers
527 views

Solving the functional Equation $f(f(x))=f(x)+x$

Let $f$ be continuous on $\mathbb{R}$. Then how to find all continuous functions satisfying $f(f(x))=f(x)+x$
9
votes
2answers
2k 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).
8
votes
1answer
454 views

Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$

Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every ...
6
votes
1answer
534 views

Proving that Ring of Complex Entire functions is neither Artinian nor Noetherian

Question: To prove that the Ring of Complex Entire functions is neither Artinian nor noetherian. Proof: Clearly $R$ is not Artinian because it is a commutative integral domain which is not a field, ...
6
votes
2answers
466 views

Maximal Ideals in the Ring of Complex Entire Functions

Let $X = \mathcal{C}([0,1],\mathbb{R})$, be the ring of all continuous valued functions $f:[0,1] \to \mathbb{R}$. For $x \in [0,1]$, let $M_{x} = \{ f \in M \ | \ f(x)=0\}$. One can show by using ...
0
votes
1answer
303 views

Vector spaces with no Complete norms

Can anyone give me an example of a vector space $V$ such that there is no norm which is complete on $V$?
13
votes
3answers
940 views

Expressing $\sin(2x)$ as a polynomial of $\sin{x}$

Using trignometric identities ( double angle forumlas) one can see that $\cos{2x} = 2 \: \cos^{2}{x} - 1$ can be expressed as a polynomial of $\cos{x}$, where $p(\cos{x})=2 \: \cos^{2}{x}-1$. Then its ...
7
votes
5answers
1k views

Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$

Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$ I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then ...