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Stokes Phenomenon
In complex analysis the Stokes phenomenon, discovered by , is that the asymptotic behavior of functions can differ in different regions of the complex plane, and that these differences can be described in a quantitative way. These regions are bounded by what are called Stokes lines or anti-Stokes lines. Stokes lines and anti-Stokes lines Somewhat confusingly, mathematicians and physicists use the terms "Stokes line" and "anti-Stokes line" in opposite ways. The lines originally studied by Stokes are what some mathematicians call anti-Stokes lines and what physicists call Stokes lines. (These terms are also used in optics for the unrelated Stokes lines and anti-Stokes lines in Raman scattering). This article uses the physicist's convention, which is historically more accurate and seems to be becoming more common among mathematicians. recommends the term "principal curve" for (physicist's) anti-Stokes lines. Informally the anti-Stokes lines are roughly where some term in the asymp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f(x) \sim g(x) \qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stokes Line
__NOTOC__ Stokes shift is the difference (in energy, wavenumber or frequency units) between positions of the band maxima of the absorption and emission spectra (fluorescence and Raman being two examples) of the same electronic transition. It is named after Irish physicist George Gabriel Stokes. Sometimes Stokes shifts are given in wavelength units, but this is less meaningful than energy, wavenumber or frequency units because it depends on the absorption wavelength. For instance, a 50 nm Stokes shift from absorption at 300 nm is larger in terms of energy than a 50 nm Stokes shift from absorption at 600 nm. When a system (be it a molecule or atom) absorbs a photon, it gains energy and enters an excited state. One way for the system to relax is to emit a photon, thus losing its energy (another method would be the loss of energy as translational mode energy (via vibrational-translational or electronic-translational collisional processes with other atoms or molecules)). When the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anti-Stokes Line
__NOTOC__ Stokes shift is the difference (in energy, wavenumber or frequency units) between positions of the band maxima of the absorption and emission spectra (fluorescence and Raman being two examples) of the same electronic transition. It is named after Irish physicist George Gabriel Stokes. Sometimes Stokes shifts are given in wavelength units, but this is less meaningful than energy, wavenumber or frequency units because it depends on the absorption wavelength. For instance, a 50 nm Stokes shift from absorption at 300 nm is larger in terms of energy than a 50 nm Stokes shift from absorption at 600 nm. When a system (be it a molecule or atom) absorbs a photon, it gains energy and enters an excited state. One way for the system to relax is to emit a photon, thus losing its energy (another method would be the loss of energy as translational mode energy (via vibrational-translational or electronic-translational collisional processes with other atoms or molecules)). When the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raman Scattering
Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by a molecule as incident photons from a visible laser are shifted to lower energy. This is called normal Stokes Raman scattering. The effect is exploited by chemists and physicists to gain information about materials for a variety of purposes by performing various forms of Raman spectroscopy. Many other variants of Raman spectroscopy allow rotational energy to be examined (if gas samples are used) and electronic energy levels may be examined if an X-ray source is used in addition to other possibilities. More complex techniques involving pulsed lasers, multiple laser beams and so on are known. Light has a certain probability of being scattered by a material. When photons are scattered, most of them are elastically scattered (Rayleigh scatt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue. Formal description Consider an open subset U of the complex plane \mathbb. Let a be an element of U, and f\colon U\setminus\\to \mathbb a holomorphic function. The point a is called an ''essential singularity'' of the function f if the singularity is neither a pole nor a removable singularity. For example, the function f(z)=e^ has an essential singularity at z=0. Alternative descriptions Let \;a\; be a complex number, assume that f(z) is not defined at \;a\; but is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Branch Cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation ''w''2 = ''z'' for ''w'' as a function of ''z''. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Properties Every entire function can be represented as a power series f(z) = \sum_^\infty a_n z^n that converges everywhere in the complex plane, hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Functions
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] |
Argument (complex Analysis)
In mathematics (particularly in complex analysis), the argument of a complex number ''z'', denoted arg(''z''), is the angle between the positive real axis and the line joining the origin and ''z'', represented as a point in the complex plane, shown as \varphi in Figure 1. It is a multi-valued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg ''z'') is used. It is often chosen to be the unique value of the argument that lies within the interval . Definition An argument of the complex number , denoted , is defined in two equivalent ways: #Geometrically, in the complex plane, as the 2D polar angle \varphi from the positive real axis to the vector representing . The numeric value is given by the angle in radians, and is positive if measured counterclockwise. #Algebraically, as any real quantity \varphi such that z = r (\cos \varphi + i \sin \varphi) = r e^ for some positive real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
