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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An idea involving Cartesian products

While reading Analysis-I by Terence Tao, I came across the notion of tuples, which are objects of a Cartesian product. In one example he writes that the empty Cartesian product is not {}, but a singleton set {()}, in which () is described is the…
Siddhu
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find which function corresponds to $\displaystyle{\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}}$

I want to find which function corresponds to $\displaystyle{\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}}$. That's what I have tried: $$a_{2k-1}=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a_{2k}=\frac{(-1)^k}{(2k)!}, \ \ \ \ \ \ k\in…
evinda
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Instead of defining a function, could I do it in an other way?

I am looking at the exponential power series: $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$ It is $R=\displaystyle{\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{|a_n|}}}=\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{n!}}=+\infty$ So,the power series converges at…
evinda
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Fourier transforms and convolutions

I am really stuck at problems relating to this subject. This leads me to believe that I have fundamentally misunderstood some properties or something. I have two examples that I would like to have clarified. The first one is: "Compute the Fourier…
KryptoJon
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How to find the closure of the following set?

How to find the closure in $l^{\infty}$ of the set $$M =\{(a_1,a_2,..........) : \textrm{all but finitely many } a_i=0\}?$$
andres
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reason why the series does not converge

Consider the series $$\sum_{n=1}^{\infty} e^{-nx}$$ Why do we conclude that the series does not converge,if $x \leq 0$ ? Because of the fact that $f_n(x)=e^{-nx} \to +\infty, \text{ if }x \leq 0 $ ?
evinda
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Riemann-Stieltjes integral

Compute the following Riemann-Stieltjes integral: The integral from 0 to 3 of $dg(x)/(sqrt(1 + x^2))$ where $g(x) = |2x - 2| + |3x - 6| - |x - 3|$ What I have tried: Using formula: sum of $f(c_i)*(g(x_(i+1)) - g(x-i))$. --> $1/\sqrt(2)*(1 - 5) +…
lemonhead
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large possible regions where $\tan z , \sin z , \cos z$ are injective

$$\tan z = \frac{\sin(z)}{\cos(z)}, \cos(z), \sin (z) \ \forall z \in \mathbb{C}$$ Injectivity means: $f(x)=f(y) \Rightarrow x=y$ so i think all intervals of the form $[0,u< 2\pi]$ are good. And I also think this is wrong. Dose somebody see how…
VVV
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A function such that $f(x) > x f'(x)$.

Any help? I thought about solving the differential equation $f(x) = x f'(x)$ but I don't think that does anything useful.
Brian
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Expand Rouche Theorem region by maximum modulus of dominating part

What I am attempting to show (constructively) That if the point which maps to the maximum modulus on the contour $\partial{K}$ is known then the region $K$ can be expanded in the neighborhood of that point so that a region $K'$ obtains $\ni$…
AutoIncrement
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Elementary problems

Find $n$ such that $n$ is a positive integer satisfying the following equation. $$2(2^2)+3(2^3)+4(2^4)+\ldots+n(2^n)=2^{n+10}$$ Can anybody help ? I can't believe this is an elementary school problem...
Vendrik Kho
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Why do we pick $n_0$ such that $\frac{1}{n_0}< \delta$?

Let $f: \mathbb{R} \to \mathbb{R}$ uniformly continuous.We set $f_n(x)=f(x+\frac{1}{n})$.Show that $f_n \to f \text{ uniformly }$. Let $\epsilon>0$. Since $f: \mathbb{R} \to \mathbb{R}$ uniformly continuous, $\exists \delta>0 $ such that $\forall…
evinda
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Show that $x_n \rightarrow 0$

Let $f:[0,1] \rightarrow \mathbb{R}$ continuous, such that $f(0)=0$ We set $x_n=\int_0^1{f(x^n)}dx$ Show that $x_n \rightarrow 0$ $$$$ The function $f$ is continuous at a closed interval $\Rightarrow $ $f$ is bounded $\Rightarrow \exists M>0: |f(x)|…
Mary Star
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Radius of convergence | ratio test

I need to find the radius of convergence of $\Sigma n^3z^n$ I want to use the ratio test because it would be simpler than the root test. If $C_n=n^3$ then $| \dfrac {C_{n+1}}{C_n}| > 1$ because $(n+1)^3 > n^3$ for $\forall n>0$. The problem I face…
blubberbrot
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a Fourier transform (sinc)

let $K(u) = \frac{\sin(u)}{\pi u}$ show that Fourier transform of $K$ is $ \hat{K}(\omega) = \textbf{1}_{|\omega|\leq 1} $ Some help would be appreciated
John
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