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Zernike Polynomials
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and imaging. Definitions There are even and odd Zernike polynomials. The even Zernike polynomials are defined as :Z^_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \! (even function over the azimuthal angle \varphi), and the odd Zernike polynomials are defined as :Z^_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \! (odd function over the azimuthal angle \varphi) where ''m'' and ''n'' are nonnegative integers with ''n ≥ m ≥ 0'' (''m'' = 0 for spherical Zernike polynomials), ''\varphi'' is the azimuthal angle, ''ρ'' is the radial distance 0\le\rho\le 1, and R^m_n are the radial polynomials defined below. Zernike polynomials have the property of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zernike Polynomials With Read-blue Cmap
Frits Zernike (; 16 July 1888 – 10 March 1966) was a Dutch physicist who received the Nobel Prize in Physics in 1953 for his invention of the phase-contrast microscope. Early life and education Frederick "Frits" Zernike was born on 16 July 1888 in Amsterdam, Netherlands to Carl Friedrich August Zernike and Antje Dieperink. Both parents were teachers of mathematics, and he especially shared his father's passion for physics. In 1905 he enrolled at the University of Amsterdam, studying chemistry (his major), mathematics and physics. Academic career In 1912, he was awarded a prize for his work on opalescence in gases. In 1913, he became assistant to Jacobus Kapteyn at the astronomical laboratory of Groningen University. In 1914, Zernike and Leonard Ornstein were jointly responsible for the derivation of the Ornstein–Zernike equation in critical-point theory. In 1915, he became lector in theoretical mechanics and mathematical physics at the same university and in 1920 he was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergeometric Function
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by and . There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic. History The term "hypergeometric series" was first used by John Wallis in his 1655 book ''Arithmetica Infinitor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi. The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Neumann
Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German Mathematical physics, mathematical physicist and professor at several German universities. His work focused on applications of potential theory to physics and mathematics. He contributed to the mathematical formalization of Electromagnetism, electrodynamics and analytical mechanics. Neumann boundary condition, Neumann boundary conditions and the Neumann series are named after him. Biography Carl Gottfried Neumann was born in Königsberg, Province of Prussia, Prussia, as one of the four children of the mineralogist, physicist and mathematician Franz Ernst Neumann (1798–1895), who was professor of mineralogy and physics at the University of Königsberg. His mother Luise Florentine Hagen (born 1800) was the sister-in-law of mathematician Friedrich Wilhelm Bessel. Carl Neumann is brother of Ernst Christian Neumann, a German physician. Carl Neumann studied primary, secondary and university studies in K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Trigonometric Functions
The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral. Generally, if the function \sin x is any trigonometric function, and \cos x is its derivative, \int a\cos nx\,dx = \frac\sin nx+C In all formulas the constant ''a'' is assumed to be nonzero, and ''C'' denotes the constant of integration. Integrands involving only sine * \int\sin ax\,dx = -\frac\cos ax+C * \int\sin^2 \,dx = \frac - \frac \sin 2ax +C= \frac - \frac \sin ax\cos ax +C * \int\sin^3 \,dx = \frac - \frac +C * \int x\sin^2 \,dx = \frac - \frac \sin 2ax - \frac \cos 2ax +C * \int x^2\sin^2 \,dx = \frac - \left( \frac - \frac \right) \sin 2ax - \frac \cos 2ax +C * \int x\sin ax\,dx = \fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rodrigues' Formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. describes the history of the Rodrigues formula in detail. Statement Let (P_n(x))_^\infty be a sequence of orthogonal polynomials on the interval , b/math> with respect to weight function w(x). That is, they have degrees deg(P_n) = n, satisfy the orthogonality condition \int_a^b P_m(x) P_n(x) w(x) \, dx = K_n \delta_ where K_n are nonzero constants depending on n, and \delta_ is the Kronecker delta. The interval , b/math> may be infinite in one or both ends. More abstractly, this can be viewed through Sturm–Liouville theory. Define an operator Lf := - \frac (Wf')', then the differential eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James C
James may refer to: People * James (given name) * James (surname) * James (musician), aka Faruq Mahfuz Anam James, (born 1964), Bollywood musician * James, brother of Jesus * King James (other), various kings named James * Prince James (other) * Saint James (other) Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Film and television * James (2005 film), ''James'' (2005 film), a Bollywood film * James (2008 film), ''James'' (2008 film), an Irish short film * James (2022 film), ''James'' (2022 film), an Indian Kannada-language film * "James", a television Adventure Time (season 5)#ep42, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign Function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as \sgn x or \sgn (x). Definition The signum function of a real number x is a piecewise function which is defined as follows: \sgn x :=\begin -1 & \text x 0. \end The law of trichotomy states that every real number must be positive, negative or zero. The signum function denotes which unique category a number falls into by mapping it to one of the values , or which can then be used in mathematical expressions or further calculations. For example: \begin \sgn(2) &=& +1\,, \\ \sgn(\pi) &=& +1\,, \\ \sgn(-8) &=& -1\,, \\ \sgn(-\frac) &=& -1\,, \\ \sgn(0) &=& 0\,. \end Basic properties Any real number can be expressed as the product of its absolute value and its sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Photolithography
Photolithography (also known as optical lithography) is a process used in the manufacturing of integrated circuits. It involves using light to transfer a pattern onto a substrate, typically a silicon wafer. The process begins with a photosensitive material, called a photoresist, being applied to the substrate. A photomask that contains the desired pattern is then placed over the photoresist. Light is shone through the photomask, exposing the photoresist in certain areas. The exposed areas undergo a chemical change, making them either soluble or insoluble in a developer solution. After development, the pattern is transferred onto the substrate through etching, chemical vapor deposition, or ion implantation processes. Ultraviolet, Ultraviolet (UV) light is typically used. Photolithography processes can be classified according to the type of light used, including ultraviolet lithography, deep ultraviolet lithography, extreme ultraviolet lithography, extreme ultraviolet lithography ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American National Standards Institute
The American National Standards Institute (ANSI ) is a private nonprofit organization that oversees the development of voluntary consensus standards for products, services, processes, systems, and personnel in the United States. The organization also coordinates U.S. standards with international standards so that American products can be used worldwide. ANSI accredits standards that are developed by representatives of other standards organizations, government agencies, consumer groups, companies, and others. These standards ensure that the characteristics and performance of products are consistent, that people use the same definitions and terms, and that products are tested the same way. ANSI also accredits organizations that carry out product or personnel certification in accordance with requirements defined in international standards. The organization's headquarters are in Washington, D.C. ANSI's operations office is located in New York City. The ANSI annual operating ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Optical Society
Optica, founded as the Optical Society of America (later the Optical Society), is a professional society of individuals and companies with an interest in optics and photonics. It publishes journals, organizes conferences and exhibitions, and carries out charitable activities. History Optica was founded in 1916 as the Optical Society of America, under the leadership of Perley G. Nutting, with 30 optical scientists and instrument makers based in Rochester, New York. It soon published its first journal of research results and established an annual meeting. The group's ''Journal of the Optical Society of America'' was created in 1918. The first series of joint meetings with the American Physical Society took place in 1918. In 2008, it changed its name to the Optical Society. In September 2021, the organization's name changed to Optica, in reference to the organization's journal by the same name and geographic neutrality to reflect the society's global membership. In 2024, followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
