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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sum of negative integer of zeta function

What is the Sum of negative integers of zeta function? $$ \sum_{s=1}^{\infty} \zeta(-s) = \zeta(-1) + \zeta(-2) + \zeta(-3) + \zeta(-4) + ... $$ The negative even number of the zeta function is zero. and the negative odd number appears…
Sukhyun Park
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Proving that a function is not totally differentiable in $(0,0)$

I am trying to show that $g(x,y)=y f(x,y)$ with $$f: \mathbb R^2 \rightarrow \mathbb R $$ $$ f(x,y) = \begin{cases} \dfrac {2xy^2}{x^2+y^4} & (x,y)\ne (0,0) \\\\ 0 & (x,y)=(0,0) ~ \end{cases} $$ is not totally differentiable in $(0,0)$. What I…
B.Swan
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How to show this attribute for regular integrals?

I need to show this attribute for any $ x \in \mathbb{R}$ for regular integrals: $$\int_0^x f(t)(x-t) \, dt = \int_0^x \left( \int_0^u f(t) \,dt \right) \,du$$ Well,my plan was to solve each of the sites..and show that they are equal.But that doesnt…
wondering1123
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Minimum over a convex set

Suppose $Z$ is a closed convex set and $x\not\in Z$. We need to show that there exists $z'\in Z$ such that $||z'-x||\leq||z-x||~~\forall z\in Z$. I know that the distance function is continuous hence has a minimum on any compact subset of $Z$ but…
QED
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formula for f*$\omega$

I have the following problem: Let $U,V\subset\mathbb{R}^n$ be open subsets, $f:U \rightarrow V$ a diffeomorphism and $\omega \in \Omega^nV$. Find a formula for $f^*\omega$. My problem is that I don't know the meaning of $f^*\omega$. $\Omega^nV$ is…
Tobi92sr
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Sum of $\sin(kx)/k = \pi-x/2$

i need to show this equality: $$ \sum_{k=1}^n \frac{\sin(kx)}{k} = \frac{\pi - x}{2}$$ I should use that $\displaystyle\frac{\sin(kx)}{k} = \int_\pi ^x \cos(kt)\,\mathrm dt$. I tried many times to solve this, but I just got stuck. Is there a…
wondering1123
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Proving some sup over integrals is finite (could we exchange $\sup$ and $\int$?).

Could someone help me to prove that $$I=\sup_x \int_{|x-y|<1}|q(y)|^2 \omega(x-y) \, dy <\infty$$ where $x,y\in \mathbb{R}^n$, $q(y)$ is bounded for all $y$ and $\omega(x-y)$ is a function given by: $\mathbf{i})$ negative powers of $|x-y|$, e.g.,…
R. Ferreira
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Shuffled Convergent Sequence

Question: Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence ($x_1 ,y_1 ,x_2 ,y_2 ,...,x_n ,y_n…
James Snell
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proving one lim with multiplied terms is two lims of the terms multiplied

The problem is as follows: Let c be a cluster point of A subset R and suppose that f: A -> R and g: A -> R are functions such that the limits of f(x) and g(x) exist as x goes to c. Prove that: $$lim(f(x)g(x)) = (limf(x))(lim(g(x))$$ as x goes to…
Adagio
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Does this proof need to be more rigorously proven?

So I have a problem where I am to determine the limit or prove it doesn't exist. The problem is: $$ \lim_{x\to 0} \sin(x)\cos\left(\frac{1}{x}\right) $$ My proof is as follows (I'm pretty confident about this): For all $ x \ne 0, -1 \le…
Sky
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Is this metric space incomplete?

Possible Duplicate: which of the following metric spaces are complete? I have doubt to this problem., $X=(0,\pi/2)$ and the metric is $d(x,y)=|\tan x-\tan y|$ is it complete metric space? I took one example $x_n=\frac{1}{2^n}$ is a cauchy…
Myshkin
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Is it possible to solve this equation using the Lambert function?

I am trying to solve the following equation, where $0 \leq a < b \leq 1$ are constants and $x \in (a,b)$; $$\frac{x-a}{b-a} = e^{-2\log(2)/(x+1)}$$ and stumbled across the Lambert W-function which I can use if I can transform my equation into…
malin
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Showing a metric space is bounded.

This is from a review packet: Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$ i) Show that $(\mathbb{R},d)$ is a bounded metric space. ii) Show that $A=[a,\infty)$ is a closed and bounded subset of…
emka
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Proving limit of a cauchy sequence

{x$_n$} is a cauchy sequence such that for every N in natural numbers there exists m,n >= N such that x$_m$ < 0 and x$_n$ > 0. Prove lim x$_n$ = 0. Attempt: Since it's cauchy, it is convergent, say lim x$_n$ = L. Suppose L > 0, then x$_m$ > 0 for…
Adagio
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Does there exist a variation of Minkowski's inequality with differences instead of sum.

I seek something of the form $\|f-g\|_p \leq |\|f\|_p-\|g\|_p|$.
Chao Xu
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