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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Prove a series that equals to $\frac{e}{e-1}$

Prove that $$ \lim_{N\to\infty}\sum_{k=0}^\infty \left(1+\frac{k}{N}\right)^{-N}=\frac{e}{e-1} $$ I think $\sum_{k=0}^\infty \left(1+\frac{k}{N}\right)^{-N}$ should be a Riemann sum of a function but could find it. What is the trick in this…
Gatsby
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2D basis functions orthogonal under exponential kernel

In one dimension, the Laguerre polynomials are orthogonal under exponential weighting: $$ \int_0^\infty L_n(x) L_m(x) e^{-x} \, dx = 0, n \ne m $$ Does anyone know what the corresponding basis functions would be in 2…
Lou Scheffer
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Find the distance between two poles based on length of sagging string between them

A string is drawn between two 10 meter tall poles. At its lowest point, the string is 3 meters off the ground. The total length of the string is 14 meters. What's the distance between the two poles?
Itamaram
  • 133
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Analysis Problem

Given that: For any $[a, b]\subset (-\infty,+\infty)$, $f$ is integrable in $[a,b]$, $p>0$, and ${\mid f\mid}^{p}$ is integrable in $(-\infty,+\infty)$. Prove that $$\lim_{h\to0}\int_{-\infty}^{+\infty }\mid {f(x+h)-f(x)}\mid ^{p} dx=0 $$
mse
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Equivalence of norms on the space of smooth functions

Let $E, F$ be Banach spaces, $A$ be an open set in $E$ and $C^2(A,F)$ be the space of all functions $f:A\to F,$ which are twice continuously differentiable and bounded with all derivatives. The question is when following two norms in $C^2(A,F)$ are…
Georgii Riabov
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Rank theorem implies inverse function theorem

I am studying analysis on $\mathbb{R}^n$ and there is this question I cannot solve. Indeed it was not asked to me in any sense, but is usual to hear people saying that rank theorem is also one of the equivalents theorem to the inverse function…
L.F. Cavenaghi
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Prove that a subset C of $\mathbb R^n$ is closed if and only if it contains all its limit points.

Prove that a subset C of $\mathbb R^n$ is closed if and only if it contains all its limit points. A closed set is defined by a set of all boundary points My professor said "We may prove that C is not closed if and only if $C^c$ has a limit point…
user320317
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Does there exist a continuous function whose composition with itself is the exponential map?

All of the maps $$ F(x) = x^4 \\ G(x) = \exp (\exp x) \\ H(x) = \sin (\sin x) $$ can be expressed as the self-compositions of the functions $$ f(x) = x^2 \\ g(x) = \exp x \\ h(x) = \sin x $$ So this led me naturally to the question whether other…
eepperly16
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Is periodic extension of Lipschitz function Lipschitz?

Let $f: [0,T] \rightarrow \mathbb{R}$, where $T>0$, be a Lipschitz with constant $K$ and $f(0)=f(T)$. Let us define $g(x)=f(x)$ for $x \in [0,T]$ and $g(x+T)=g(x)$ for $x \in \mathbb{R}$. Does $g$ satisfies $$|g(x)-g(y)| \leq K |x-y|$$ for $x,y…
Alex
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If $n\in N$ and $f(x)=\ln(1+x^{2n})$, then derivative $f^{(2n)}(-1)=0$.

If $n\in N$ and $f(x)=\ln(1+x^{2n})$, then derivative $f^{(2n)}(-1)=0$. I try: $e^{f(x)}=1+x^{2n}$ ,$(f'e^f)'=f''e^f+(f')^2e^f$ but I don't know what next.
piteer
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Is Thomae's function Riemann integrable?

Let $\displaystyle f: [0,1] \rightarrow \mathbb{R}$ given by $$f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \\ 0 & x = 0 \\ \\ \frac{1}{q_x} & x = \frac{p_x}{q_x} \in \mathbb{Q} \backslash \{0\}, \ p_x \in \mathbb{Z}, \ q_x \in \mathbb{N},…
ghshtalt
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Equicontinuous, bounded, and closed implies pointwise compact

I'm trying to prove a particular version of the Arzela-Ascoli theorem. I have already gone through the general version: Let $A \subset M$ where $A$ is compact and $M$ is a metric space. Let $B \subset C_{b}(A,N)$ where $N$ is a metric space and…
user308485
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How can I prove that the argument of a transcendental function must be dimensionless?

We all know from school that arguments of transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. But is there a simple way to prove it?
emanuele
  • 369
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Proving $\sup\left\{ r\in\mathbb{Q}:r^{2}<3\right\}=\sqrt{3}$

Let $E=\left\{ r\in\mathbb{Q}:r^{2}<3\right\}$. Prove that $\sup E=\sqrt{3}$. Since $E$ is bounded from above by $\sqrt{3}$ and is nonempty, $\alpha:=\sup E$ must exist by the Least Upper Bound Principle. Now I am stuck. Suppose…
Galois
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Prove that a continuous function on (0,1) has a sequence of step functions which converge pointwise to it on [0,1]

Suppose $f:\left[0,1\right]\rightarrow R$ is continuous on $\left(0,1\right)$. Prove there is a sequence of step functions $\left\{f_{n}\right\}$ which converge pointwise to $f$ on $\left[0,1\right]$. John Franks - A (Terse) Introduction to…
duqu
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