Dirac Delta Function Integral Over Finite Interval
I am taking a course on Partial Differential Equations at the moment. We were introduced to the Dirac Delta function, and were told it satisfies$$\int^b_a \delta(x)=1$$ for $a<0<b$.Having taken a...
View Articlewhat they mean by asymptotic to?
I don't understand this please who understand ?I have read somewhere that: As $j$ tends to $\infty$, $\displaystyle\sum_{j=0}^{n-1} j^{a}$ is asymptotic to $cn^{a+2}$ if $a>-1$, is asymptotic to...
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Can I interchange differentiation and integration on this contour integral?
I would like to obtain an integral representation for $2\operatorname{Ai}^2(z)\operatorname{Ai}'(z)$. Is it permissible to interchange differentiation and integration on the known...
View ArticleProve that $\frac{1}{n+1} \leq \sin(\frac{1}{n})$ for all $n\geq1$.
I need to use the following inequality in a proof: $\frac{1}{n+1} \leq \sin(\frac{1}{n})$ for all $n\geq1$. However, I don't know how I can justify or prove that this inequality holds. Any ideas?
View Article$f(x): f'(x) = 4x^3[f(x)]^2, \forall x \in \mathbb{R}$ and $f(2) =...
$f(x): f'(x) = 4x^3[f(x)]^2, \forall x \in \mathbb{R}$ and $f(2) = -\dfrac{1}{25}.$ Find $f(1).$In the solution, they divide both sides by $[f(x)]^2$ to get $\dfrac{f'(x)}{[f(x)]^2} = 4x^3.$ Then both...
View ArticleTesting the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges. One can then ask the question of...
View ArticleOn the contrary, a theorem of Rudin's real and complex analysis [closed]
A theorem of Rudin's real and complex analysissuppose $\mu$ and $\nu_1$,$\nu_2$ are measures on a $\sigma$_algebra m, and $\mu$ is positive:if $\nu_1<<\mu$ and $\nu_2\perp\mu$, then...
View ArticleHow to show conservation of mass for the heat equation?
I have a question about a property of the solutions to the heat equation. Let $u(t,x)$ be a solution to the (one-space dimension) heat equation $$u_t = u_{xx}.$$ Is it true that...
View ArticleLong time behaviour of the integral of the solution to the heat equation
I am interested at longtime behaviour of the solution to the one space dimension heat equation. That is, the solution to the equation $$u_{t} = \frac{1}{2}u_{xx},$$with initial condition $u(0,x)$ which...
View ArticleAn example for converse of a theorem in Rudin
A theorem of Rudin's real and complex analysissuppose $\mu$ and $\nu_1$,$\nu_2$ are measures on a $\sigma$_algebra m, and $\mu$ is positive:if $\nu_1<<\mu$ and $\nu_2\perp\mu$, then...
View Article$f\in C[0,1]\cap C^1(0,1)$ with $f'$ bounded, but $f$ not differentiable at...
I'm looking for an elementary example of a function $f$ satisfying$f$ is continuous on $[0,1]$, differentiable on $(0,1)$,The derivative $f'$ is bounded on $(0,1)$,However, the function $f$ is not...
View ArticleDoes $\forall x(x\in B\implies x\in A)$ imply that $((\forall a\in...
I was solving a problem like$$\textrm{if} \:f\in C(A)\:\text{and}\:B\subset A,\: \text{then}\: f|_B\in C(B).$$ I transformed $f\in C(A)$ so $f$ is continuous on A and therefore$$(\forall a\in...
View ArticleWhat's wrong with my solution of James stewart 1.7 Q 36?
Question - Prove that $\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$My Solution $|x - 2| < \delta$$|\frac{1}{x} - \frac{1}{2}| < \epsilon$$\frac{\delta}{|2(\delta+2)|}<\epsilon\qquad$ (as...
View ArticleAsymptotic behaviour of $\displaystyle \int_0^\infty f_1(x+z)f_2(x)dx$ in...
Let $f_1$ and $f_2$ be two positive continuous functions defined on $\mathbb{R}_+$ and $a_1,a2>0$ such that:$$f_1(x)\sim_{+\infty} e^{-a_1 x}\quad\mathrm{ and } \quad f_2(x)\sim_{+\infty} e^{-a_2...
View ArticleConvergence of Fourier series in $L_2$ [closed]
Let $f \in L_2(\mathbb{R})$ and $S_n$ is the $n-th$ partial Fourier sum. prove that $\lim_{n \to \infty} \int_{-\pi}^{\pi}{|f(x) - S_n(x)|dx} = 0$I guess we need to use the fact that trigonometric...
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About adjoint of densely defined linear operator
Let's consider $\mathfrak{H}$ be a Hilbert space, a densely defined linear operator on $\mathfrak{H}$ is an ordered pair $(T,D_T)$, where $D_T$ is a dense linear subspace of $\mathfrak{h}$, and...
View ArticleIf f is differentiable on [a,b] then it is Lipschitz of order 1?
I understand how to prove the weaker case where you add the hypothesis that f is continuously differentiable on [a,b] and slightly less weaker , when f’ is bounded. However, to disprove the statement I...
View ArticleThe Fourier transform of $e^{-i/x}$
$\def\R{\mathbb R}$Question.Does anyone know what is the Fourier transform of$$ f(x)=e^{-i/x} $$on the real line? I would like to compute it explicitly, or to establish some properties to have a good...
View ArticleSeeking Algorithm to Solve a Convolution Integral or Directly Convolve Two CDFs
Convolution of CDFs in Polynomial FormHi everyone,I'm working on a project where I need to find a way to directly convolve two cumulative distribution functions (CDFs) given in polynomial form, and...
View ArticleUniform convergence of exponential series
Using the definition of uniform convergence prove that the exponential series $\sum_{k=0} ^\infty \frac{x^k}{k!}$ converges uniformly on any finite subinterval of $\mathbb{R}$.The sequence of functions...
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