integralnyintegran

Integralnyintegran is a term used in some pedagogical discussions to describe the integrand that defines a given integral transform or integral equation. In this usage, the focus is on the function inside the integral sign and how it varies with the integration parameter and any external parameters that influence the transform. The concept emphasizes the role of the integrand as a primary object, alongside the limits and the kernel that governs the transformation.

Definition. Consider a domain D and an integral of the form I(x) = ∫_a^b K(x,t) f(t) dt, where K is a kernel and f is a base function with appropriate integrability. For each x in D, the inner expression g_x(t) = K(x,t) f(t) is the integrand associated with x. The collection of these inner expressions {g_x} over x ∈ D constitutes the integralnyintegran in this framework. The term highlights how the integrand depends on x through the kernel and the base function.

Properties. The integralnyintegran inherits linearity in f: if f1 and f2 are integrable and α, β are scalars, then the integrand corresponding to α f1 + β f2 is α g^1_x + β g^2_x. If K is smooth and f is integrable, the resulting I(x) often inherits smoothness in x. Variations in a, b, or the form of K alter the shape of the integralnyintegran without changing its fundamental role as the object under the integral.

Example. Let a = 0, b = 1, K(x,t) = x t, f(t) = t. Then g_x(t) = x t^2 and I(x) = ∫_0^1 x t^2 dt = x/3.

Applications. The concept is used to illustrate kernel methods, Volterra or Fredholm-type integral equations, and the analysis of how changes in the integrand affect the transformed outcome.

See also. Integrand, kernel function, integral transform, Volterra equation.