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
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Power series of $\frac{z}{e^z-1}$

I have the function $ f(z)=\frac{z}{e^z-1} $. I'm looking for a power series. My idea: I define a power series $$ \frac{z}{e^z-1}=f(z):=\sum\limits_{k=0}^\infty c_k\cdot z^k. $$ Now I start equating the coefficients $ c_k $: $$…
hallo007
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Prove U = $\mathbb{R}^2$\ {(0,0)} is open in $\mathbb{R}^2$

I have to prove that $U = \mathbb{R}^2\setminus \{(0,0)\}$ is open in $\mathbb{R}^2$. I know that $U$ is open if $U = \operatorname{inw}(U)$ (collection of all interior points of $U$) (this is the Dutch notation, not sure about the English one). I…
myrthe
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Show that there is a $ x \in [0,1]$ that, $f(x) = g(x)$

Let $f, g: [0, 1] → [0, 1]$ be continuous functions such that $f(0)\leqslant g(0)$ and $f(1) \geqslant g(1)$. Show that there is $x ∈ [0, 1]$ such that $f(x) = g(x)$. Is it enought to draw a grap? Or i need to show something more?
anna
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Check if a student will pass an MCQ exam with negative marking

Here is a scenario: We have an MCQ exam with 100 questions. Every 3 wrong answers will remove 1 correct answer. The Minimum needed score to pass this exam is 50 I'm using the following formula : score = ( (correct * 3) - wrong) / (3 * total)) *…
far2005
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Ultrametric completion does not introduce new norms

I could not solve this problem: Prove that for a non-Archimedian field $K$ with completion $L$, $$\left\{|x|\in\mathbb R \mid x\in K\right\} =\left\{|x|\in\mathbb R \mid x\in L\right\}$$ I considered a Cauchy sequence in $K$ with norms having…
quanta
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How can I prove $f(x) = \frac32 x − \frac23$ is surjective?

So I'm trying to prove the function $f(x) = \frac32 x − \frac23$ is bijective (injective and surjective)... For injection I can easily write $\frac32x - \frac23 = \frac32y - \frac23$; that gives me $x=y$ so it's injective. So the one more step to…
Minotaur666
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Prove that $f(x)=x^3$ is Riemann integrable on $[0,1]$ using partitions

Consider the function $f(x)=x^3$. Prove it is Riemann integrable on $[0,1]$ using partitions.
theoxoxo
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sectionally/piecewise continuous functions

Assume $f$ and $g$ are two piecewise continuous functions on an interval $( a , b )$ containing the point $t_0$ . Assume further that $f$ has a jump discontinuity at $t_0$ while $g$ is continuous at $t_0$ .How can i verify that the jump in the…
salv.t
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Can anyone tell me whether this is errata or not? "Analysis 1" by Terence Tao

In the Proposition 2.1.16, I guess that the sentence "Then we can assign a unique natural number a(n) to each natural number n, such that a(0)=c and a(n++)=f(an) for each natural number n" should be like this : "Then we can assign a unique natural…
ju so
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Asking for an example of a (number) system failed to exist

Quite a few analysis books I've looked into start with some number system, claim some (reasonably convincing) axioms on it, assume it to be existed, and build analysis from there. Some explain why the number system exists by constructing it from…
J-A-S
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Prove that if $x, y ∈$ B̅$(z,r)$ such that $||x - y|| ≥ εr$, then $||z - {{x+y} \over 2}|| ≤ r \sqrt{1 - {{ε^2} \over 4}}$.

Let $z ∈ ℝ^n$, $r > 0$, and $ε ∈ (0, 2]$. Prove that if $x, y ∈$ B̅$(z,r)$ such that $||x - y|| ≥ εr$, then $||z - {{x+y} \over 2}|| ≤ r \sqrt{1 - {{ε^2} \over 4}}$. my take on this exercise (not much, but I tried): Knowing that B̅$(z,r)$ is a…
just1frustredstudent
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Integral equations

Are there functions $f(z)$ that satisfy the following: $$\int_1^\infty \frac{\mathrm{d}z}{f(z)} = \frac{1}{\int_{1}^{\infty} f(z) \, \mathrm{d}z} $$
yomcflow420
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Convergence of flows of uniformly converging vector fields

My questions is about the flows of certain "well behaved" sequence of vector fields. Let $X^n: U \subset R^n \rightarrow R^n$ be a sequence of smooth vector fields which i. Are equicontinous and equibounded ii. $X^n$(x) converges fastly and…
Sina
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Dieudonné's proof of Taylor's formula

I'm having trouble understanding a lemma which Dieudonné in "Foundations of Modern Analysis", chapter 8.14, pp. 185-186, uses to proof Taylor's formula, and then the proof of the formula itself with this lemma. First, for proving the lemma,…
Roland Salz
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proving the infimum of a set

I am trying to find the supremum and infimum of the set $A = \{1/x +(-1)^x|x \in \mathbb{N} \}$. Obviously, the supremum is $3/2$, however, I am not sure about the infimum. In the limit as $x \rightarrow \infty$, $1/n \rightarrow 0$ and since…
smejak
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