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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Difference between geometric transformation and substitution in integral

In my first semester of university I learned using substitution for integrals (I study physics not math). Now I took a math class where we were introduced to transformation, for example polar coordinates, etc. I already knew polar coodinates from…
Lizzi
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Possible generalization of averages

I was thinking if it's possible to generalize the idea of averages, like we have already the arithmetic, geometric and harmonic averages and I realized that they share some properties and I'd like to know if there's a theory about averages and…
eipi10
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prove $\inf S = -\sup (-S)$

Let $-S$ be the set $\left\{ -s:s \in S \right\}$ where $-S$ is the set that contains negatives of the members of $S$. We want to prove that $\inf(S) = -\sup(-S)$ Here is how I proved it Let $s_0= \sup(-S)$. That is for all $-s_1\in -S$ then $-s_1…
user60887
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What does it mean for a sequence $x_k \in E$ such that if $x_k \to x$, then $f(x_k) \to f(x)$?

This is from a homework problem, but this is not itself the homework problem. I'm trying to understand what everything means. Please be patient as I'm an on and off again student and I just keep feeling hopelessly lost and too embarrassed to talk…
Jason
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Show that $f:\mathbb{R}^2 \rightarrow \mathbb{R}, f(x,y)=x^2+y$ is continuous in $\mathbb{R}^2$ by $\epsilon - \delta$ definition of continuity.

I need to show that $f:\mathbb{R}^2 \rightarrow \mathbb{R}, f(x,y)=x^2+y$ is continuous in $\mathbb{R}^2$ (everywhere) by $\epsilon - \delta$ definition of continuity. First, I write the definition. $f$ continuous $\Leftrightarrow \forall…
matholympian
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if A is compact and B is closed, there exist $u$ in $A$ and $v$ in $B$ such that $\|u-v\|=\inf\{\|a-b\|\mid a\text{ in }A\text{ and }b\text{ in }B\}$

I can't prove this theorem. Could you help me? Thanks for your helps. $A$ and $B$ are subsets of an $n$-dimensional Euclidean space. Prove that, if $A$ is compact and $B$ is closed, there exist $u$ in $A$ and $v$ in $B$ such that…
matholympian
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Existence of solution of equation continuous $f(x)=0$

$f:[0,+\infty)\rightarrow \mathbb{R}$ is continuous function. s.t $f(0)=1$, $f(1)=0$ then existence of minimal $x>0$ s.t $f(x)=0$ is guaranteed? If it has counterexample, for $C_1$ function $f$, is guaranteed the existence of minimal $x>0$ which…
백주상
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How to construct a function with the following property?

I am reading a paper where the author states that a map with the following property exists. Let $\Omega\subset B(0,R)\subset \mathbb{R}^d$ for $R>0$ and $n\geq 1.$ Define $p_n$ to be a function that goes from $\Omega$ to a grid of at most $(2Rn)^d$…
Student
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Baby Rudin 1.6 a. Alternate Proof

The proof I got for 1.6 a. in Baby Rudin is different from those I've seen elsewhere. I wonder if I made a mistake? 1.6: Fix $>1$. Prove that if $,,,$ are integers, $>0,>0$ and $=\frac{}{}=\frac{}{}$, then $(^)^{\frac{1}{}}=(^)^{\frac{1}{}}$ My…
lightnesscaster
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Describe all functions which are defined as $f:\mathbb R \to \mathbb R $ and satisfying $f(x)=\frac1{f(x)}$

Describe all functions which are defined as $f:\mathbb R \to \mathbb R $ and satisfying $$f(x)=\dfrac1{f(x)}$$ My Attempt: If $f$ is constant function then. $$(f(x))^2=1$$ Then $f(x)=1$ or $f(x)=-1$ I cannot think non-constant function satisfying…
Jale'de jaled
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Limit of trigonometric function (cotangent)

I need to solve the following problem: $\lim_{x\to 3}(x-3) \cot{\pi x}$. Can anyone give me a hint? I have no idea.
Ilja
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Proving $\lim_{x\to\infty} \frac{\Gamma(x+a)}{x^a\Gamma(x)}=1$

For any real $a>0$: $$\lim_{x\to\infty} \frac{\Gamma(x+a)}{x^a\Gamma(x)}=1$$ I got the hint to use Stirling's approximation but I cant seem to solve it. Any tips would be very appreciated.
Ereboss
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corollary to the completeness axiom

The corollary states "Every nonempty subset $S$ of $\mathbb{R}$ that is bounded below has a greatest lower bound inf S. The part I don't get in the proof is from where they came up with the set $-S$ where $-S=(-s : s\in S)$? Did they create it or…
user60887
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Condition for interchanging infinite sum with improper integration

As we know, uniform convergence on a finite interval can lead to the interchange of infinite sum with integration. However, it seems that for improper integration, stronger conditions are needed. For example, if $f_n(x)$ uniformly converges to…
ZWJ
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