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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217 views

On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$f(s)=-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log ...
3
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75 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
3
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48 views

Notation of an infinite integral

I'm confused regarding the notation of the integral I've underlined in green. Does this mean that the integral over any range is $\infty$?
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68 views

Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
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50 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
3
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54 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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35 views

Evaluating a triple integral explained step by step

Evaluate: $$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where $D$ is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ...
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32 views

How could i find the pdf of exponential distribution from its characteristics function?

I know that the characteristics function of the exponential distribution is as following: $$ \phi_x(t) =\frac{\lambda}{(\lambda -it)}$$ Also, I know that the pdf of the exponential distribution is: ...
3
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106 views

Integral vs antiderivative

I have a similar question to this one: Integrable or antiderivative. If a function has an antiderivative, does the difference of values of the antiderivative on the endpoints of an interval always ...
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40 views

If an integral over the plane vanishes, prove that it vanishes on a square.

Let $f\in L^1(\mathbb{R}^2)$ with respect to the Lebesgue measure $m\times m$ on $\mathbb{R}^2$. Prove that if $$\iint_{\mathbb{R}^2} f(x,y)dxdy=0$$ then there exists a square $S_{a,b}=\{(x,y)\,|\, ...
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35 views

How do I tackle this integral: $\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$? Is my solution correct?

I want to solve the following integral: $$\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$$ I did the following: Substitute $\gamma(k) = k-k_0 \Leftrightarrow k = \gamma + k_0;~\gamma(\pm\infty) = ...
3
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41 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
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49 views

Simplifying a Fourier integral

I have what is effectively a Fourier integral resulting from applying Abbe's theorem that I would like to simplify (ideally into a closed form solution): $$ f(\theta_0,\theta_1;\alpha) = ...
3
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46 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
3
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75 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
3
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84 views

Prove the converse of convolution theorem

I am trying to prove the converse of convolution theorem: $$ \mathscr{F}[f(x)g(x)]=\frac{1}{\sqrt{2\pi}}\,\widetilde{f}(\omega)*\widetilde{g}(\omega)$$ I try to apply the definition of convolution ...
3
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103 views

An integral identity for $\frac{x^{a-1}}{x^b-1}$ via. partial fractions

Can somebody please confirm or correct the following? If $a$ and $b$ are both positive integers such that $a<b$ and $b$ is even then we can write ...
3
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54 views

Definite integral including the Chebyshev polynomial

I would like to know the proof of $$ \int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N $$ where $T_n(x)$ is the Chebyshev polynomial of ...
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338 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
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113 views

An interesting integral

How to integrate: $$ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}.$$ I read about this problem on the Wikipedia Risch algorithm page, they gave an answer but I am at a loss how they got it....
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102 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
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65 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
3
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79 views

Solving exponential integral

Any idea how to solve this integral, I tried the integration by parts, and it made the things even more difficult. The substitution didn't work either. Here is the integral: $\displaystyle \int ...
3
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116 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
3
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37 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
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74 views

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 ...
3
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58 views

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 ...
3
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78 views

Two properties about Bessel function

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 ...
3
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313 views

Determine the indefinite integral: $I=\int \frac{\sin^5x\arcsin^5x}{e^{5x}\ln^5x}dx$

Determine the indefinite integral: $$ I = \int{\sin^{5}\left(x\right)\arcsin^{5}\left(x\right) \over {\rm e}^{5x}\ln^{5}\left(x\right)}\,{\rm d}x $$ I have not seen this ever, so do not know how ...
3
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168 views

Determine the indefinite integral: $I=\int \frac{\ln x}{x\sqrt{1-4\sin x-\ln^2x}}dx$

Determine the indefinite integral: $$I=\int\frac{\ln x}{x\sqrt{1-4\sin x-\ln^2x}}dx$$ Really I don't know how, hence please help me. I need a solution!
3
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92 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ ...
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114 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
3
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71 views

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 ...
3
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106 views

Integrals for children

I was just reviewing a chapter on integration defined through step functions, and was wondering how would you explain the concept of an integral of a step function to a child ?
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128 views

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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44 views

infinite sum and integral representation

assuming all the numbers $ k_{n} $ are REAL is then true that $$ \sum_{n=0}^{\infty}\frac{1}{k_{n}^{s}}=\frac{1}{2 \Gamma (s) cos(\pi s)}\int_{0}^{\infty}dtt^{s-1}\sum_{n=0}^{\infty}cos(k_{n}t) $$ ...
3
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0answers
123 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
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110 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
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51 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
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80 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
3
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0answers
68 views

How to calculate hard integral?

How to calculate the integral $$ \int_D \frac {\prod_{i<j}(a_i-a_j)^2\prod_{i<j}(b_i-b_j)^2} {\prod_{i,j} (a_i+b_j)^2}\,d\lambda_{2n-1},$$ where $$D:=\{ (a_1,\dots ...
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55 views

Can Fredholm integral equation of the first type be represented as a differential equation?

Can Fredholm integral equation of the first type be represented as a differential equation? In other words, given a Fredholm integral equation of the second type does there exist a differential ...
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143 views

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 ...
3
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60 views

Difficult Definite Integral with inverse Cosh

How can I solve this integral containing inverse cosh? Does it have any antiderivative? $$ \int_b^r t^2 \operatorname{arccosh}(a/t) \sqrt{r^2 - t^2} d t$$ for $0< b< r< a$.
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57 views

Vanishing of integral over hemispheres implies vanishing of function?

Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that A) The integral of $F$ over the hemisphere $(x-x_0)^2 + (y-y_0)^2 + z^2 = R^2$ ...
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294 views

Integrating a fractional power of a rational function

I am currently working on a project where I stumbled upon the integral $$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$ where $R$ is a ...
3
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0answers
97 views

Double Integration.

I have an integral $$\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y})~dx~dy~d\bar{x}~d\bar{y}$$ where $f= ...
3
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0answers
152 views

Integral of a max function over a hypersphere; expected max z score

Is there a closed-form expression for the integral of max($x_1,..., x_n$) over the ($n-2$)-dimensional hypersphere, {$x \in \mathbf{R}^n$: $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i^2 = 1$}? I come ...
3
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204 views

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 ...
3
votes
0answers
81 views

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 ...