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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...

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what 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...

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Prove 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?

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$f(x): f'(x) = 4x^3[f(x)]^2, \forall x \in \mathbb{R}$ 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...

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Testing 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...

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On 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...

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How 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...

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Long 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...

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An 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...

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$f\in C[0,1]\cap C^1(0,1)$ 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...

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Does $\forall x(x\in B\implies x\in A)$ 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...

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What'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...

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Asymptotic 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...

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Convergence 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...

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If 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...

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The 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...

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Seeking 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...

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Uniform 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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