Encyclopedia - Bessel Function: Encyclopedia - Bessel Function
In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonic...   » Read the article

Encyclopedia - Bessel Function: Encyclopedia Ii - Bessel Function - Definitions
Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, h...   » Read the article

Encyclopedia - Angular Resolution: Encyclopedia - Angular Resolution
Angular resolution describes the resolving power of a telescope. Angular resolution - Definition of terms. Resolving power is the abili...   » Read the article

Encyclopedia - Airy Disc: Encyclopedia - Airy Disc
Due to the wave nature of light, light passing through apertures is diffracted, and the diffraction increases with decreasing aperture si...   » Read the article

Encyclopedia - Airy Disc: Encyclopedia Ii - Airy Disc - Mathematical Details
The intensity of the Fraunhofer diffraction pattern of a circular aperture is given by: where J1 is a Bessel function of the first kin...   » Read the article

Encyclopedia - Bc Programming Language: Encyclopedia Ii - Bc Programming Language - Posix Bc
The POSIX standardised bc language is traditionally written as a program in the dc programming language to provide a higher level of acce...   » Read the article

Encyclopedia - Gnu Scientific Library: Encyclopedia Ii - Gnu Scientific Library - Example
The following example program calculates the value of the Bessel function for 5 [1]: #include <stdio.h> #include <gsl/gsl_sf_be...   » Read the article

Encyclopedia - Helmholtz Equation: Encyclopedia Ii - Helmholtz Equation - Solving The Helmholtz Equation Using Separation Of Variables
The general solution to the spatial Helmholtz equation can be obtained using separation of variables. In spherical polar coordinates, ...   » Read the article

Encyclopedia - List Of Letters Used In Mathematics And Science: Encyclopedia Ii - List Of Letters Used In Mathematics And Science - Latin
Ã… - Ã…ngström, A - Ampere, Area, a Blood type, a Spectral type, Vector potential, Work, B - B meson, a Blood type, Boron, Luminance, Ma...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Overview
Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph of the delt...   » Read the article

Encyclopedia - Aliasing: Encyclopedia Ii - Aliasing - Mathematical Explanation Of Aliasing
The preceding explanation and the Nyquist criterion are somewhat idealised, because they assume instantaneous sampling and other slightly...   » Read the article

Encyclopedia - Angular Resolution: Encyclopedia Ii - Angular Resolution - Explanation
The resolving power of a lens is ultimately limited by diffraction effects. The lens' aperture is a "hole" that is analogous to a two-dim...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Derivatives Of The Delta Function
The derivative of the Dirac delta function (also called a doublet) is the distribution δ' defined by for every test function . From t...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Fourier Transform
The continuous Fourier transform of the Dirac delta is the constant function . The inverse transform of this constant function will be th...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Delta Function Of More Complicated Arguments
A helpful identity is the scaling property: and so This concept may be generalized to: where xi are the roots of g(x). In the in...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Equivalent Definition
The Dirac delta function is a distribution δ(ξ) whose indefinite integral is the function usually called the Heaviside step functio...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Representations Of The Delta Function
The delta function can be viewed as the limit of a sequence of functions where δa(x) is sometimes called a nascent delta function. Th...   » Read the article

Encyclopedia - Angular Resolution: Encyclopedia Ii - Angular Resolution - Microscope Case
The resolution D depends on the angular aperture α: . Here α is the collecting angle of the lens, which depends on the width of objec...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Derivatives Of The Delta Function
The derivative of the Dirac delta is the distribution δ' defined by for every test function . From this it follows that xδ'(x) = âˆ...   » Read the article

Encyclopedia - Angular Resolution: Encyclopedia Ii - Angular Resolution - Telescope Case
Point-like sources separated by an angle smaller than the angular resolution cannot be resolved. A single optical telescope has an angula...   » Read the article

Encyclopedia - Dirac Delta Function: Encyclopedia Ii - Dirac Delta Function - Formal Introduction
The Dirac delta is often introduced with the property: valid for any continuous function f. However, there is no actual function δ(x)...   » Read the article

Encyclopedia - Aliasing: Encyclopedia Ii - Aliasing - Overview
Aliasing - Aliasing in periodic phenomena. The sun moves east to west in the sky, with 24 hours between sunrises. If one were to take a...   » Read the article

Encyclopedia - List Of Letters Used In Mathematics And Science: Encyclopedia Ii - List Of Letters Used In Mathematics And Science - Greek
Î’ - Beta function, Γ - Angular momentum, Gamma function, Christoffel symbols, Δ - A macroscopic change in the value of a variable, Lap...   » Read the article

Encyclopedia - Bc Programming Language: Encyclopedia Ii - Bc Programming Language - Example Code
Since the bc ^ operator only allows an integer power to its right, one of the first functions a bc user might write is a power function w...   » Read the article

Encyclopedia - Aliasing: Encyclopedia Ii - Aliasing - Caveats
It is extremely important to keep in mind what is much repeated in the above discussion: the Nyquist theorem, the optimality of the sinc ...   » Read the article

Encyclopedia - Bc Programming Language: Encyclopedia Ii - Bc Programming Language - Gnu Bc
GNU bc derives from the POSIX standard and includes many enhancements. It is entirely separate from dc-based implementations of the POSIX...   » Read the article

Encyclopedia - Bessel Function: Encyclopedia Ii - Bessel Function - Asymptotic Forms
The Bessel functions have the following asymptotic forms. For small arguments 0 < x << 1, one obtains: where α is non-neg...   » Read the article

Encyclopedia - Bessel Function: Encyclopedia Ii - Bessel Function - Applications
Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical co...   » Read the article