Leibniz integral rule

Résumé

In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form \int_{a(x)}^{b(x)} f(x,t),dt, where -\infty < a(x), b(x) < \infty and the integrands are functions dependent on x, the derivative of this integral is expressible as \frac{d}{dx} \left (\int_{a(x)}^{b(x)} f(x,t),dt \right ) = f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) ,dt where the partial derivative \tfrac{\partial}{\partial x} indicates that inside the integral, only the variation of f(x, t) with x is considered in taking the derivative. It is named after Gottfried Leibniz. In the special case where the functions a(x) and b(x) are constants a(x)=a and b(x)=b w

Source officielle

https://en.wikipedia.org/wiki/Leibniz_integral_rule

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The weighted averages method for semi-infinite range integrals involving products of Bessel functions

Ruzica Golubovic, Juan Ramon Mosig

An efficient and accurate method, based on the weighted averages (WA) extrapolation technique, is presented for the evaluation of semi-infinite range integrals involving products of Bessel functions of arbitrary order. The method requires splitting the integration interval into a finite and an infinite part. The integral over the first finite part is computed using an adaptive quadrature rule based on Patterson formulas. For the evaluation of the remaining integral, the strongly irregular oscillatory behavior of the product of two Bessel functions is first represented as a sum of two asymptotically simply oscillating functions. Then, by applying the integration-then-summation technique, a sequence of partial integrals is obtained, and its convergence is accelerated with the help of WA. Details and possible complications involved in the method are addressed. Finally, the excellent performance of the proposed method is demonstrated throughout several numerical examples.

Institute of Electrical and Electronics Engineers2013

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A critically degenerate elliptic Dirichlet problem, spectral theory and bifurcation

Charles Stuart

In an open, bounded subset Omega of R-N such that 0 is an element of Omega we consider the nonlinear eigenvalue problem -Sigma(N)(i,j,=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + V(x)u + n(x,del u)+ g(x, u) = lambda u in Omega integral(Omega) u(2) + Sigma(N)(i,j=1) A(ij)partial derivative(j)u partial derivative(i)u dx < infinity and u = 0 on partial derivative Omega, where V is an element of L-infinity(Omega) and the nonlinear terms n and g are of higher order near 0 so that the formal linearization about the trivial solution u 0 is - Sigma(N)(i,j=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + Vu = lambda u. The leading term is degenerate elliptic on Omega because it is assumed that there are constants C-2 >= C-1 > 0 such that C1 vertical bar x vertical bar(2)vertical bar xi vertical bar(2)

2020

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