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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
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Result relating to Stirling's Formula

Have a question that I'm stuck on here. Let $$r_n= \frac{\sqrt{n}}{n!}\left(\frac{n}{e}\right)^n$$ Express $\log\left(r_{n+1}/r_n\right)$ as simply as possible. For this I got $\left(\frac{1}{2}+n\right)\log\left(1+\frac{1}{n}\right)-1$. b) Verify…
user65972
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Is reflection or rotation in a $2$-dimensional normed space isometric?

Is reflection in the $x$-axis or in the line $y=x$ in a $2$-dimensional normed space isometric? How about rotation through a right angle? If so, what is the proof?
LoveMath
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Completing a proof

Say we are given this: Impossibility of ordering the complex numbers. As yet we have not defined a relation of the form $x < y$ if $x$ and $y$ are arbitrary complex numbers, for the reason that it is impossible to give a definition of $<$ for…
Carlo Sanchez
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bonus question on quaternions

I have this asterisk question, I know its hard to do and I know no one would get it in my class. Just wondering if any of you guys could give me good hints in how to do this. It would be appreciated. Thank you Question The quaternions extend the…
MathGeek
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A problem from Zorich

I am trying to read Mathematical Analysis I by Zorich on my own. Here is an exercise that I could not solve: For all $l \in \mathbb{R}$ that is not of the form $\frac{1}{n}$ for some $n \in \mathbb{N}$, there exists a continuous function $f: [0,…
Ziyang Guo
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Find a value of $h$ such that if $\left | x \right |< h$ then $\sin(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+R$ where $\left | R \right |< 10^{-4}$.

I need to find a value of $h$ such that if $\left | x \right |< h$ then $$\sin(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+R$$ where $\left | R \right |< 10^{-4}$. Attempt: I tried to start with $\left | \sin(x) \right |\leq 1$, then using triangle…
user4167
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Integrate real valued function from complex integral

I'm having difficulties as to evaluating the complex integral. Also, how should I use its result to evaluate the real valued integral ? Integrate the function $$f(z) = \frac{z}{1 - a e^{-i z}} ;\: a > 1$$ around a suitable rectangle to…
engineeringstudent
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How to show that given $\epsilon\in(0,1)~\exists~\delta\in(0,1)$ such that $\delta*\delta*\delta<\epsilon?$

Let $*:[0,1]\times[0,1]\to[0,1]$ be a continuous $t$-norm i.e. a) $*$ is continuous, b) $*$ is commutative and associative, c) $1*a=a~\forall~a\in[0,1],$ d) $a\le b,c\le d\implies a*c\le b*d.$ How to show that given…
Jave
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An exercise from Zorich's book (edit and add a) and b))

This exercise come from Zorich,Mathematical Analysis,I P232 Exercise 6 6 Let $f \in C^{(n)} ( ]-1,1[ )$ and $\sup_{-1
Laura
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Equivalent definition of measure zero in a Manifold.

In Munkres's analysis on manifold, it defines: Let $M$ be a compact k-manifold in $\mathbb{R^n}$, of class $C^r$. A subset $D$ of $M$ said to have measure zero in $M$ if it can be covered by countably many coordinate patches $\alpha_i:…
user533661
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Use Weierstrass to show $f(x)=0$

I was reviewing for a test and I'm not sure how to approach this problem: Suppose $f$ is a function continuous on $[0,1]$ with $$\int_0^1x^nf(x)dx=0$$ for all even integers $n\geq0$. Then $f(x)=0$ for all $x\in[0,1]$. Is this true? And if so, how…
analysisq
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Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$

Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$? (I don't understand what is meant by this question)
Jhwana
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$\frac{\int fg dx}{\int g dx}=f(0)$

We already know that $$\lim_{n \rightarrow +\infty} \int_{-1}^1 (1-x^2)^n dx = 0$$ If we have $f(x) \in C[-1,1]$ then prove $$\lim_{n \rightarrow +\infty} \frac{\int_{-1}^1 f(x)(1-x^2)^n dx}{\int_{-1}^1 (1-x^2)^n dx } = f(0)$$ My thought is…
XT Chen
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Rational Function Theorem related to Integration?

I know how to use this algorithm when I am integrating rational functions, but my textbook has omitted the actual proof for why it works. If someone could please help me with this question:
Dick
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Show an $L^2$ function is $L^1$

I am trying to show that for all $f \in L^2([0,2\pi])$, we also have $f \in L^1([0,2\pi])$ by Cauchy-Schwarz. I really couldn't see how Cauchy-Schwarz could be applied here. If I apply it to $(f,f)$, I still would have an $L^2$ norm. Can you give…
Alvis Nordkovich
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