Scorer's Function

picture info	Scorer's Function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the Scorer's functions are special functions studied by and denoted Gi(''x'') and Hi(''x''). Hi(''x'') and -Gi(''x'') solve the equation :y''(x) - x\ y(x) = \frac and are given by :\mathrm(x) = \frac \int_0^\infty \sin\left(\frac + xt\right)\, dt, :\mathrm(x) = \frac \int_0^\infty \exp\left(-\frac + xt\right)\, dt. The Scorer's functions can also be defined in terms of Airy functions: :\begin \mathrm(x) &= \mathrm(x) \int_x^\infty \mathrm(t) \, dt + \mathrm(x) \int_0^x \mathrm(t) \, dt, \\ \mathrm(x) &= \mathrm(x) \int_^x \mathrm(t) \, dt - \mathrm(x) \int_^x \mathrm(t) \, dt. \end File:Plot of the Scorer function Gi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Special Function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Airy Function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solutions to the differential equation \frac - xy = 0 , known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). Definitions For real values of ''x'', the Airy function of the first kind can be defined by the improper Riemann integral: \operatorname(x) = \dfrac\int_0^\infty\cos\left(\dfrac + xt\right)\, dt\equiv \dfrac \lim_ \int_0^b \cos\left(\dfrac + xt\right)\, dt, which converges by Dirichlet's test. For any real number x there is positive real number M such that function \dfrac3 + xt is increasing, unbounded and convex with continuous and unbounded derivative on interval lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]