Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

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).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
2
votes
0 answers

show that f is integrable at $[a,c]$ and $[c,b]$

Let $f:[a,b] \to \mathbb{R}$ bounded and $c \in (a,b)$.Then $f$ is integrable at $[a,b]$ iff $f$ is integrable at $[a,c]$ and $[c,b]$.In this case,we have $\int_a^b f = \int_a^c f + \int_c^b f$. The proof for the direction $\Rightarrow$ is like…
evinda
  • 7,823
2
votes
3 answers

Some problem about Cauchy sequence.

I cannot solve this: Let X be a complete metric space with a metric d. (a) Suppose that the sequence $x_{n}$ in X satisfies $\sum_{n=0}^{\infty}d(x_{n},x_{n+1})<\infty$ Show that $x_{n}$ converges (b) Suppose that X is nonempty and f is a function…
user143299
  • 31
2
votes
2 answers

differential forms, stokes theorem in higher dimension

The problem is: Consider the differential form $a=p_1dq_1+p_2dq_2-p_1p_2dt$ in the space of $R^5$ with coordinates $(p_1,p_2,q_1,q_2,t)$. (a) compute $da$ and $da\wedge da$ (b) Evaluate the integral $\int_S t da\wedge da$ where $S$ is the 4-dim…
breezeintopl
  • 1,437
2
votes
0 answers

Is all the nonelementary function that is not piecewise could be express as infinite series of elemetary function?

Are all the non-elementary functions that is not piecewise expressible as an infinite series of elementary functions? details about elementary function - http://en.wikipedia.org/wiki/Elementary_function
Victor
  • 8,372
2
votes
1 answer

Proof of $\left |\frac{\sin(n+1/2)t}{\sin{t/2}}-\frac{\sin{nt}}{\tan{t/2}}\right| \leq 1$

I need help to proof $$\left |\frac{\sin(n+1/2)t}{\sin{t/2}}-\frac{\sin{nt}}{\tan{t/2}}\right| \leq 1$$
Niklas Hebe
  • 111
2
votes
2 answers

Verifying if a function is Lipschitz

Let $\Omega\subset \mathbb{R}^N$ be a domain with $N\ge 2$. Let $K\subset \Omega$ be a compact set and take $u:\overline{\Omega}\to\mathbb{R}$ such that $u$ is Lipschitz and $u=1$ in $\partial K$. Assume that $\operatorname{int}{K}\ne\emptyset$ and…
Tomás
  • 22,559
2
votes
1 answer

integrate over a contour

Please help me with this: Find the value of $\displaystyle\int_{\gamma}\frac{e^z}{z-Logz}dz$, $\gamma$ is the positively oriented contour consisting of four vertices at $2, 4, 4+3i, 2+3i$
Joyce
  • 361
2
votes
1 answer

Homeomorphism on the unit circle

Can somebody tell me how to prove that $f:[0,2π)→S^1$ given by $t↦⟨\cos t,\sin t⟩$, where $S^1$ is the unit circle in the plane, and $[0,2π)$ is the real interval, 1. is continuous at the point 0 (the Problem I am facing consists on the fact that if…
ivo
  • 21
2
votes
1 answer

Why is a set countable if there is a injective function?

For school, I have to prove that every finite subset of $\mathbb N$ is countable. Wikipedia tells me, that "By definition a set $S$ is countable if there exists an injective function $f$ from $S$ to the natural numbers.". I'm probably missing…
fabian789
  • 171
2
votes
3 answers

The function is continuous but not uniformly continuous at $[0,1) \cup (1,2]$.

I want to show that the function $$f=\left\{\begin{matrix} 0, \text{ if } x \in [0,1)\\ 1, \text{ if } x \in (1,2] \end{matrix}\right.$$ is continuous but not uniformly continuous at $[0,1) \cup (1,2]$. A function $f:A \rightarrow \mathbb{R}$ is…
Mary Star
  • 13,956
2
votes
1 answer

Could we use also use $\delta_1,\delta_2$,or do we have to use the same $\delta$?

I am looking at the following exercise: $f,g:A \to R$ are uniformly continuous at $A$.If we suppose that $f,g$ are bounded,prove that $f \cdot g:A \to R$ is uniformly continuous at $A$. I have thought to do it like that: Let $\epsilon'>0$.Since $f$…
evinda
  • 7,823
2
votes
3 answers

Analysis (Absolute Value )

The question is let $a \in \mathbb{R} $ does not contain 0. Prove that $|a+\frac{1}{a}| \ge 2$. I have no idea how to start this problem and any help on it would be greatly appreciated.
user129509
  • 89
2
votes
2 answers

Prove that the set of all algebraic numbers is countable.

I'm a student in Korea. If I make a mistake in grammar, please indicate. Recently, I'm studying the book 'Principles of Mathematical Analysis' So, I tried to solve the exercise #2 in chapter 2. 'A complex number $z$ is said to be algebraic if there…
user128766
  • 1,077
2
votes
1 answer

The Baire space $\mathscr{N}$ is separable

Given is the Baire space $\mathscr{N}$. The elements are functions (or sequences) $f : \mathbb{N} \to \mathbb{N}$ and the metric $d$ is given by $d(f, g) = \frac{1}{k}$ if $f(i) = g(i)$ for all $1 \leq i \leq k-1$ and $f(k) \neq g(k)$. Problem: show…
Math Student 020
  • 1,599
2
votes
1 answer

Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, $\lim_{n \to \infty}na_{n}=0$.

It is given that $a_{n}$ is a positive and decreasing sequence. Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, $\lim_{n \to \infty}na_{n}=0$. That's what I tried.Could you tell me if it is right?? $\sum_{n=1}^{\infty}a_{n}$ converges,so it…
Mary Star
  • 13,956
1 2 3
…
99 100