Tagged Questions

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
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Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
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Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
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solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
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Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
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Integration of a function where the integral is equal to 0

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Suppose that $\displaystyle\int_a^b x^nf(x) \, dx=0$ for all $n\in\{ 0,1,2, \ldots \}$. Prove that $f=0$. (Hint. Consider ...
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infinite sum and integral representation

assuming all the numbers $k_{n}$ are REAL is then true that $$\sum_{n=0}^{\infty}\frac{1}{k_{n}^{s}}=\frac{1}{2 \Gamma (s) cos(\pi s)}\int_{0}^{\infty}dtt^{s-1}\sum_{n=0}^{\infty}cos(k_{n}t)$$ ...
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Powers of the Meijer G function

I am trying to find the following integral: $$I = \int_0^{+\infty} J_{1}(2\beta\sqrt s) \Psi(m,1-m,\beta^2/a)^L d\beta,$$ where $J_{1}(2\beta\sqrt s)$ is the known Bessel function, $\Psi$ is the ...
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Multivariable calculus: optimizing for shortest path along a curvy plane?

I want to write a computer program which can help me spend the least amount of energy and time walking between locations on my university campus. My campus is very hilly, and it is also extremely hot ...
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Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
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Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$\alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t).$$ Show ...
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What do I need to know to integrate any rational function?

My analysis book makes the following statement: Every rational function with real coefficients can be integrated in terms of rational functions, logarithm functions, arctangent ...
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Understand integral from Gradshteyn and Ryzhik book “Table of integrals, series, products”

I was checking useful integrals in this book. I have found one (6.298) that is what I need, but I don't understand how every step towards the final result works. ...
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spectral integral

I am learning spectral integration for my summer. I am stuck at a point. Having got hold of a spectral measure, we define the spectral integral of a simple function as usual and then approximate any ...
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Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
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Integration by parts of a normalized function - [copied from Physics.SE]

By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that ...
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Simplify the integral with error function

$\newcommand{\erf}{\operatorname{erf}}$ I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of $$\erf(\infty)$$ I ...