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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved ...
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives .
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
The Euler–Cauchy stress principle states that upon any surface (real or imaginary) that divides the body, the action of one part of the body on the other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body, [2] and it is represented by a field (), called the traction vector, defined on ...
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
This estimate follows from Cauchy's integral formula (in the general form) applied to = where is a smooth function that is = on a neighborhood of and whose support is contained in . Indeed, shrinking U {\displaystyle U} , assume U {\displaystyle U} is bounded and the boundary of it is piecewise-smooth.
According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e. ′ = which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of J [ f ] {\displaystyle J[f]} and is denoted δ J {\displaystyle \delta J} or δ f ( x ...