Calculate The Sum Of The Infinite Series
Given$$\begin{aligned}\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{x}{2})^{2n-\frac{1}{2}}}{n! \,\Gamma(n+\frac{1}{2})} &=...
View ArticleShow that $f'$ is unbounded on $(0,1]$ but that $f$ is nevertheless uniformly...
The question is this. Let $f(x) = \sqrt{x}, \forall x \geq 0.$ Show that $f'$ is unbounded on $(0,1]$ but that $f$ is nevertheless uniformly continuous on $(0,1].$ Compare with Theorem 19.6.The theorem...
View ArticleWhat Are Some Series One Could Do a Whole Project On? [closed]
I’m in the planning process for an 8-month project on mathematical series, their theories, and proofs behind them. I want to know what kinds of series I could make a detailed thesis and paper on with...
View Article$\max_{x \in [0,1]} |B_k(x)| = k! 2^{1-k} \pi^{-k}(1-3^{-k} + O(4^{-k}))$
How can we show that$$\max_{x \in [0,1]} |B_k(x)| = k! 2^{1-k} \pi^{-k}(1-3^{-k} + O(4^{-k})),$$where $B_k(x)$ is the $k$-th Bernoulli polynomial and $k$ is an odd integer $\geq 3$?From the Fourier...
View ArticleMistakes taking Fourier Transforms on finite time intervals
I am trying to find the all the following Fourier Transforms on a finite time interval $[t_0,\ t_f]$ with $t_0<t_f$:$$A(iw) = \int\limits_{t_0}^{t_f} f(t)\ \delta(t-t_0)\ e^{-iwt} dt$$$$B(iw) =...
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Real Analysis - Relationship between pointwise convergence of functions and...
I'm reading Sheldon Axler's book on Measure Theory. Here is the start of the proof of a theorem. I don't understand why formula 2.86 is true. I know that it comes from the definition of pointwise...
View ArticleDense subset of $L^1(X)$
Let $(X,\mu)$ be a $\sigma$-finite mearsure space. Suppose $E_1\subset E_2\subset \cdots$ is a sequence of subsets with finite measure such that $X= \cup E_i$.Let $A = \{ f\in L^1(X): f \text{ is...
View ArticleHow to show that a function is strictly decreasing.
The function $f \colon \mathbb{R} \to \mathbb{R}$ is defined by $f(x)=\frac{x^2+2x}{x^2-1}$.How would you show that $f(x)$ is a strictly decreasing function.
View ArticleA property of the upper contour set of convex functions?
We know the upper contour set of a convex function is not necessarily convex.However, suppose $f\left(y\right)>f\left(x\right)$ and $f\left(z\right)>f\left(x\right)$, without loss of generality,...
View Articleboundary term in the integration by parts formula for difference quotient
I have a sequence of values in $\mathbb{R}$ say $f_{k}$ where $k\in{0,1,\ldots, N}$ and a step $h=\frac{T}{N}$. Then I define the piecewise constant function\begin{align}f_{h}(x)=f_{k}\mbox{ for all...
View ArticleSimple proof of weak version of Markov brothers' inequality
For a polynomial $f$, write$$\|f\|=\sup_{-1\leq x\leq 1}|f(x)|.$$It is a classical result of A. Markov that $\|f'\|\leq (\deg f)^2\|f\|$.Question. Is there a simple proof of the weaker result that, for...
View Articleboundary term in the integration by parts of difference quotients
This question is related to my other question boundary term in the integration by parts formula for difference quotient.Now, let us consider two functions $f,g\in C_{0}^{\infty}(\mathbb{R})$ and a time...
View ArticleA differentiable and unbounded function with infinitely many critical points,...
I'm searching for a function $f: \Bbb R^2 \rightarrow \Bbb R$ which has following properties: Both $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are continuous in $\Bbb...
View ArticleAsymptotics for $f(z) = (z-1) \prod_n \frac{2}{z^{1/a_n}+1}$ with $1/a_1 +...
Let $Re(z) > 1$Consider functions such as$$f(z) = (z-1) ([2 / (z^{1/a_1} + 1)] [2 / (z^{1/a_2} + 1)][2 / (z^{1/a_3} + 1)][2 / (z^{1/a_4} + 1)][2 / (z^{1/a_5} + 1)]...)$$or more formally$$f(z) =...
View Articleprove that a function $f$ is uniformly continuous if and only if there exists...
Consider two metric spaces $(X,d_X)$, $(Y,d_Y)$, and a function $f: X\to Y$, $f$ is uniformly continuous.A function $w: [0,\infty)\to [0,\infty]$ is called a modulus of continuity for $f$,...
View ArticleProve there exists some $\beta < 0$ for which $W(x) \leq \beta +\sin(x)$ on...
Suppose that $W : [1,6] \rightarrow \mathbb{R}$ is continuous and $W(x) < \sin(x)$ on $[1,6].$ Prove there exists some $\beta < 0$ for which $W(x) \leq \beta + \sin(x)$ on $[1,6].$I was thinking...
View ArticleTwo representation of a particular form of hypergeometric function.
From the basic definition ofhypergeometric function,we know that$${}_{2}F_{1}(1,p;p+1;1) =p \int_{0}^{1} \dfrac{t^{p-1}}{1-t} dt. $$My professor also told me that\begin{align*}{}_{2}F_{1}(1, p;...
View ArticleShow that even and $2\pi$-periodic continutation is differentiable
Let be $f:\mathbb{R}\to\mathbb{R}$ given by an even and $2\pi$-periodic continuation of$$f(x):=\begin{cases}4x^2-\pi^2,&x\in\left[0,\frac{\pi}{2}\right]\\8\pi...
View ArticleInclusion of $L^p$ spaces
Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align}X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some $p_0>1.$I...
View ArticleDoes $\sin(n/x)$ converges a.e. to a constant?
Let us consider the sequence of functions $\sin(n/x)$, for $x\in(0,1)$.Is it true that there exists a subsequence $n_j$ such that $\sin({n_j}/x)\to 0$ a.e.?Intuitively, my guess is yes. For every...
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