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In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each.
Laplace also popularized and further confirmed Sir Isaac Newton's work. [2] In statistics, the Bayesian interpretation of probability was developed mainly by Laplace. [3] Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The following is a list of Laplace transforms for many common functions of a single variable. [1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency ).
Parabolic cylinder () function appears naturally in the Schrödinger equation for the one-dimensional quantum harmonic oscillator (a quantum particle in the oscillator potential), [+] = (), where is the reduced Planck constant, is the mass of the particle, is the coordinate of the particle, is the frequency of the oscillator, is the energy, and () is the particle's wave-function.
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase.
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in Jackson's Classical Electrodynamics. [1] Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in