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The derivative of this integral at x is defined to be | |, where |B| denotes the volume (i.e., the Lebesgue measure) of a ball B centered at x, and B → x means that the diameter of B tends to 0. The Lebesgue differentiation theorem ( Lebesgue 1910 ) states that this derivative exists and is equal to f ( x ) at almost every point x ∈ R n . [ 1 ]
The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions ...
For example, the solution to the Dirichlet problem for the unit disk in R 2 is given by the Poisson integral formula. If f {\displaystyle f} is a continuous function on the boundary ∂ D {\displaystyle \partial D} of the open unit disk D {\displaystyle D} , then the solution to the Dirichlet problem is u ( z ) {\displaystyle u(z)} given by
The result for Lebesgue measure turns out to be a special case of the following result, which is based on the Besicovitch covering theorem: if μ is any locally finite Borel measure on R n and f : R n → R is locally integrable with respect to μ, then (()) () = for μ-almost all points x ∈ R n.
An example, which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics, [61] is the rescaled Airy function / (/). Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense.
The absolute value function : +, = | |, which is not differentiable at = has a weak derivative : known as the sign function, and given by () = {>; =; < This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. For example, the definition of v(0) above could be replaced with any ...
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation.