An Etymological Dictionary of Astronomy and Astrophysics
English-French-Persian

An integral with upper and lower limits.

→ definite; → integral.

The two branches of mathematics that make up the → calculus. → differential calculus; → integral calculus.

→ differential; → integral; → calculus.

The simplest case of a → multiple integral.

→ double; → integral.

An integral used in the → Fourier transform.

→ Fourier analysis; → integral.

Two integrals that involve quadratic equations in the sine and cosine functions and are defined as: C(x) = ∫ cos (πt²/2) dt and C(y) = ∫ sin (πt²/2) dt, integrated from 0 to x. They are quite frequently used in optics studying → Fresnel diffraction and similar topics. The Fresnel integrals are also used in railway and freeway constructions. These integrals may be evaluated to arbitrary precision using → power series. Alternatively the amplitudes may be found graphically by use of → Cornu's spiral.

→ Fresnel diffraction; → integral.

Math.: An integral without upper and lower limits. The general antiderivative of a function. → definite integral.

→ definite; → integral.

1) Consisting of whole numbers or integers.
2) Mathematical function obtained by the process of → integration.

Integral, from M.Fr. intégral, from M.L. integralis "forming a whole," → integer "whole."

1) Dorostâl, from dorost "whole, complete; healthy; right," related to dorud "benediction, praise, thanksgiving," from Mid.Pers. drust "whole; healthy; well, right," drôd "health, thriving;" O.Pers. duruva- "firm, certain, immune;" Av. druua- "healthy;" cf. Skt. dhruvá- "fixed, firm, immovable, lasting, certain;" Russ. zdorovyjj "healthy;" See also → sound.
2) Dorostâl, from dorost + -âl, → -al.

Branch of the calculus that deals with integration and its use in finding volumes, areas, equations of curves, solutions of differential equations, etc.

→ integral; → calculus.

An equation involving an unknown function that appears as part of an integrand.

→ integral; → equation.

A technique in spectroscopy for recording a spectrum from each point of an extended object. The field of view image is divided into a multitude of small components using different methods, e.g. lenslet arrays, fiber bundles, or image slicers. From each component a spectrum is extracted or an image is reconstructed at a particular wavelength.

→ integral; → field; → spectroscopy.

A function whose image is a subset of the integers, i.e. that takes only integer values.

→ integral; → function.

Math.: An operator whose inverse is a differential operator.

→ integral; → operator.

The integral admitted by the equations of a body of infinitesimal mass moving under the → gravitational attractions of two massive bodies, which move in circles about their → center of gravity. The Jacobi integral is the only known conserved quantity for the circular → restricted three-body problem. In the co-rotating system it is expressed by the equation: (1/2) (x^·2 + y^·2 + z^·2) = U - C_J, where the dotted coordinates represent velocities, U is potential energy, and C_J the constant of integration (→ zero-velocity surface). The Jacobi integral has been used for two different purposes: 1) to construct surfaces of zero velocity which limit the regions of space in which the small body, under given initial conditions, can move, and 2) to derive a criterion (→ Tisserand's parameter) for re-identification of a → comet whose orbit has suffered severe perturbations by a planet. Also known as Jacobi constant.

Named after Karl Gustav Jacobi (1804-1851), a German mathematician who did important work on elliptic functions, partial differential equations, and mechanics; → integral.

A series of successive integrations in which the integral operator acts on the result of preceding integration.

→ multiple; → integral.