Evaluation of an Integral Involving the Lower Incomplete Gamma Function...
I am trying to evaluate the following integral involving the lower incomplete Gamma function raised to the power of a positive integer $N$:$\int_0^{\infty} x^{\mu-1} e^{-\beta x} \gamma(\nu, \alpha...
View ArticleSuppose $f(z)$ is analytic for $|z - z_0| > r$, then what is the series...
Suppose $f(z)$ is analytic for $|z - z_0| > r$, then do we have that $f(1/z)$ is analytic for $|z - z_0| < r$ and the following series expansion? Why?$$f(z) = \sum_{n=0}^{\infty} \frac{a_n}{(z -...
View ArticleHow to solve $\displaystyle{\lim_{x \to \infty}}{\frac{1 - e^{\frac{-1}{x +...
How to solve: $\displaystyle{\lim_{x \to \infty}}{\frac{1 - e^{\frac{-1}{x + 1}}}{1 - e^{\frac{-1}{x}}}}$?I know it's equal to 1. But how to do it rigorously? I've tried to use substitution $y =...
View ArticleHow to prove this continuous function?
let $a<b<c$. Suppose that $f$ is continuous on $[a,b]$, that $g$ is continuous on $[b,c]$, and that $f(b)=g(b)$. Define $h$ on $[a,c]$ by $h(x):=f(x)$ for $x\in[a,b]$ and $h(x):=g(x)$ for $x \in...
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Continuous functions with domain in the Natural Numbers
Can functions with domain in the Natural Numbers be continuous?In the high school, it is teached an intuitive notion of continuous functions: functions which will always appear as an "unbroken curve",...
View ArticleThe set of all points $x$ in which $f$ is continuous is $G_{\delta}$
in exercise 4.16 here: Jech - Set Theory, we are asked to prove that: Given a function $f: \mathbb{R} \rightarrow \mathbb{R}$, the set of all points $x$ in which $f$ is continuous is...
View ArticleShow $g(a)= \operatorname*{arg}_{b \in B} \, f(a,b)=0$ is continuous when...
Let $A$ and $B$ be two compact subsets of $\mathbb{R}$. Let $f:A \times B \to \mathbb{R}$ be a continuous function on $A \times B$.For each $a\in A$, define $B_a=\{b\in B:f(a,b)=0\}$, and suppose each...
View ArticleGrowth Rate From Inequality Involving Antiderivative
Let $\Omega \subset \mathbb{R}^n$, $n\geq 3$ be a bounded domain with smooth boundary, and $f\in C(\bar{\Omega}\times \mathbb{R}^1,\mathbb{R}^1)$. Suppose there exists $\mu>2$ and $M>0$ such that...
View ArticleApproximating powers of elements on the unit circle
Let $R \subseteq \mathbb{N}$. We say that $R$ is adequate if:$$\forall n \in \mathbb{N} \; \; \forall \varepsilon > 0 \; \; \forall w \in S^1 \; \; \exists r\in R \; \; |w^n - w^r| <...
View ArticleDensity property between $L^1$ and $H^1$
I would like to know if the following property is trueGiven the sobolev space $H^1(\mathbb{R}^d)$ and the Lebesgue space $L^1(\mathbb{R}^d)$, is is true that $H^1(\mathbb{R}^d)\cap H^1(\mathbb{R}^d)$...
View ArticleFind all parameters $a,b,c,d \in \mathbb R$ for which the function $f:...
My function $f: \mathbb R \rightarrow \mathbb R$ it is given a pattern:$$f(x)=\begin{cases} \frac{2^{7^{x}-1}-a}{\ln(1-x)},& x<0\\ b,& x=0 \\ \frac{\sin\left(c \sqrt{x^{2}+d^{2}x}\right)...
View ArticleFor what functions do we get back the same function we put into Newton's...
In analogy with Taylor series, we have$$ f(x + \alpha) = \sum_{n=0}^{\infty} \frac{\alpha^{\underline{n}}}{n!} \Delta^{n}f(x) = \sum_{n=0}^{\infty} \binom{\alpha}{n} \Delta^{n}f(x). $$We may take...
View ArticleA question regarding a proof about limit and continuity
I am trying to understand a part of the following proof.Prove that $(1)$ and $(2)$ are equivalent:$(1)$ $\lim_{x \to c}f(x)=f(c)$$(2)$ $f$ is continuous at $c$.I understood the proof of $(1)...
View Articlecontinuity and open subset
let say E is a subset of R and f be the real number function o E then f is continuou on E if an only if $f^{-1}(V)$ is open in E for every open subset $V$ of $R$what is opn subset V mean? in a...
View ArticleA monotonous function $f:I\rightarrow \mathbb{R}$ on an Intervall $I$ has at...
I have first tried to prove the Statement by contraposition but I don't know how I can continue on a certain step. I have also read the Solutions which I don't understand.$(x_n)$ is a sequence of not...
View ArticleIf $f_n \rightharpoonup f$ weakly in $X^{*}$, so does its convex combination?
Suppose $X$ is a reflexive normed space and let $f_n \rightharpoonup f$ weakly in $X^{*}$. Let $\{g_n\}$ be a sequence where each $g_n$ is made up of finite convex combination of $\{f_1,f_2,\ldots,...
View ArticleProof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow...
This is half of Theorem 3.19 from Baby Rudin. Rudin claims the proof is trivial. What I've come up with so far doesn't seem trivial, however, and is probably also wrong (my problem with it is pointed...
View ArticleNumber of elements in a finite $\sigma$-algebra
I have been asked to prove that the number of elements in a finite sigma algebra over a set $X$ is $2^n$ for some integer $n$. How do I go about this problem? I have no idea where to start. Thanks in...
View ArticleProving $\lim_{n \to \infty} n[n^{1/n}-1] = +\infty$
I understand that this may require the use of the more popular limit:$$\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$$but can't seem to find the correct way to use it.I've tried setting...
View ArticleUse Hölder's inequality when $\int f_ngd\mu$ is bounded
In a measure space $(X,\mathcal{M},\mu)$, say that $1<p,q<\infty$ such that $\frac{1}{p}+\frac{1}{q}=1$. Suppose $f_n\rightarrow f$ a.e., $f_n\in L^p$ for all $n$, and for each $g\in L^q$$$\sup_n...
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