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Bateman Function
In mathematics, the Bateman function (or ''k''-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by :\displaystyle k_\nu(x) = \frac\int_0^\cos(x\tan\theta-\nu\theta) \, d\theta . Bateman discovered this function, when Theodore von Kármán asked for the solution of the following differential equation which appeared in the theory of turbulence :x \frac = (x-\nu) u and Bateman found this function as one of the solutions. Bateman denoted this function as "k" function in honor of Theodore von Kármán. The Bateman function for x>0 is the related to the Confluent hypergeometric function of the second kind as follows :k_(x)=\frac U\left(-\frac\nu,0,2x\right), \quad x>0. This is not to be confused with another function of the same name which is used in Pharmacokinetics. Havelock function Complementary to the Bateman function, one may also define the Havelock function, named after Thomas Henry Havelock. In fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confluent Hypergeometric Function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harry Bateman
Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician with a specialty in differential equations of mathematical physics. With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poincare to a more expansive conformal group of spacetime leaving Maxwell's equations invariant. Moving to the US, he obtained a Ph.D. in geometry with Frank Morley and became a professor of mathematics at California Institute of Technology. There he taught fluid dynamics to students going into aerodynamics with Theodore von Karman. Bateman made a broad survey of applied differential equations in his Gibbs Lecture in 1943 titled, "The control of an elastic fluid". Biography Bateman was born in Manchester, England, on 29 May 1882. He first gained an interest in mathematics during his time at Manchester Grammar School. In his final year, he won a scholarship to Trinity College, Cambridge. Bateman studied with coach Robert Alfred Herman to prepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodore Von Kármán
Theodore von Kármán ( , May 11, 1881May 6, 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who worked in aeronautics and astronautics. He was responsible for crucial advances in aerodynamics characterizing supersonic and hypersonic airflow. The human-defined threshold of outer space is named the " Kármán line" in recognition of his work. Kármán is regarded as an outstanding aerodynamic theoretician of the 20th century. Early life Theodore von Kármán was born into a Jewish family in Budapest, then part of Austria-Hungary, as Kármán Tódor, the son of Helene (Konn or Kohn, ) and . Among his ancestors were Rabbi Judah Loew ben Bezalel, who was said to be the creator of the Golem of Prague, and Rabbi , who wrote about Zohar. His father, Mór, was a well-known educator, who reformed the Hungarian school system and founded Minta Gymnasium in Budapest. He became an influential figure and became a commissioner of the Ministry of Educa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confluent Hypergeometric Function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Henry Havelock
Sir Thomas Henry Havelock FRS (24 June 1877 – 1 August 1968) was an English applied mathematician, hydrodynamicist and mathematical physicist. He is known for Havelock's law (1907). Havelock's Law Havelock was born in Newcastle-upon-Tyne. At the age of sixteen, he entered Durham College of Physical Science. (Durham College of Physical Science was renamed Armstrong College in 1904.) He matriculated in 1897 at St John's College, Cambridge and graduated there B.A. in 1900 and M.A. in 1904. From 1903 to 1909 he was a Fellow of St John's College, Cambridge. He was a professor of applied mathematics at Armstrong College from 1914 until his retirement in 1945. (In the 1930s Armstrong College became part of King's College, Durham, which in the 1960s became part of Newcastle University.) Havelock's law Relationship between the refractive index n and the wavelength \lambda of a homogeneous material that transmits light: :k = B \ \lambda n/, where ::k = constant for the material at a giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modified Bessel Function Of The Second Kind
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separable s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Hypergeometric Functions
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