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 .

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Simply Connected domains.

If $U$ and $U'$ be two domains in $\Bbb C$, and $f$ be a homeomorphism in $U$ and $U'$ then domain $U$ is simply connected $\iff$ $U'$ is simply connected. I found this problem in complex analysis. So I would prefer to know its proof from complex…
Mat He Mat Cian
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A curious problem about Lebesgue measure.

The Problem: Let $(B(x_{m},0.5))_{m}$ be a sequence of disjoint open discs in $\mathbb{R}^{2}$ centered in $x_{m}$ and with radius 0.5. Let $\psi(n)$ be the number of these discs contained in the open disc $B(0,n)$ (that is, the disc centered in…
Katherine Rath
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What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers?

From Stephen Abbott's Understanding Analysis: Theorem: There is no rational number whose square is 2. Proof: Assume for contradiction, that there exist integers $p$ and $q$ satisfying (1) $\left(\frac{p}{q}\right)^2 = 2$. We may also assume that…
ghshtalt
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Prove that if f(x) is integrable, then so is e^(f(x)).

So here is my question: I'm working on a homework problem that deals with Jensen's Inequality. It is a rather simple application, I believe, but I'm a little stuck. Here is the problem, along with my work. Let $f$ be integrable over $[0,1]$. Show…
Calculus08
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Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that $|f(x)|^2 + |f'(x)|^2\le \max(a, b)$ for all…
Christmas Bunny
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Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda \lambda^k=-(-1)^k\frac{B_k}{k!} \textrm{for } k=1..n-2$$ The $B_k$ are the…
user11260
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How does the series $\sum_{n=1}^\infty \frac{(-1)^n \cos(n^2 x)}{n}$ behave?

The title is the question: I'm trying to understand the behavior of the series $$\star \qquad \sum_{n=1}^\infty \frac{(-1)^n \cos(n^2 x)}{n},\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad $$ where $0\leq x…
Stefan Smith
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Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I cannot do anything. Please explain. I saw this question…
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Completeness of a normed vector space

This is captured from a chapter talking about completeness of metric space in Real Analysis, Carothers, 1ed. I have been confused by two questions: What does absolutely summable mean in metric space? Does it mean the norm of xi(i=1,2,3,...) that…
Bear and bunny
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Proving $\int^{\infty}_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0$

I've been asked to prove that $$ \int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$ My approach so far has been to use a theorem proved in class that, for a random variable $X$ with characteristic…
tlonuqbar
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Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by checking $(a_n)$ is decreasing and bounded. But…
Christmas Bunny
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2013th derivative of rational function

I am struggling to find $f^{(2013)}(0)$ for $$f(x) = \frac{1}{1 + x + x^3 + x^4}$$ I know that I should use power series, and following a hint I rewrote the equation as the following: $$1 = (1 + x + x^3 + x^4)(\sum_{n = 0}^{\infty} a_n x^n) =…
lpsolver
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Convergence of a integral - heat Kernel and dirac delta function

Consider $\varphi \in S(R^n)$ (space of rapidly decreasing functions). Consider the heat kernel $$ K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, t>0 , x \in R^n$$ I want to show that…
math student
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looking for a diffeomorphism (not C1)

Let $f\colon\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ diffeomorphism with $f(B[0,1])\subset B[0,1]$ and $| \det f^{\prime}(x) |<1/2$ for all $x\in B[0,1]$ then for every continuous function $h\colon B[0,1] \rightarrow \mathbb{R}^{n}$…
Tirifilo
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Infinite Series $\sum\limits_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$

How can I find the value of the following sum? $$\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$$ $F_n$ is the Fibonacci number.($F_1=F_2=1$)
user95733
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