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Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: xy'' + (1 - x)y' + ny = 0,\ y = y(x) which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , ..., are a polynomial sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Color Plot Of The Laguerre Polynomial L N(x) With N As -1 Divided By 9 And X As Z To The Power Of 4 From -2-2i To 2+2i
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * Complex (album), ''Complex'' (album), by Montaigne, 2019, and its title track * Complex (EP), ''Complex'' (EP), by Rifle Sport, 1985 * Complex (song), "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico (rapper), Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Isotropic Harmonic Oscillator
3D, 3-D, 3d, or Three D may refer to: Science, technology, and mathematics * A three-dimensional space in mathematics Relating to three-dimensionality * 3D computer graphics, computer graphics that use a three-dimensional representation of geometric data * 3D display, a type of information display that conveys depth to the viewer * 3D film, a motion picture that gives the illusion of three-dimensional perception * 3D modeling, developing a representation of any three-dimensional surface or object * 3D printing, making a three-dimensional solid object of a shape from a digital model * 3D television, television that conveys depth perception to the viewer * 3D projection * 3D rendering * 3D scanning, making a digital representation of three-dimensional objects * 3D video game * Stereoscopy, any technique capable of recording three-dimensional visual information or creating the illusion of depth in an image * Three-dimensional space Other uses in science and technology * 3-D Sec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm Chain
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of . Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univaria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Term
In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial, :x^2 + 2x + 3,\ The number 3 is a constant term. After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial :ax^2+bx+c,\ where x is the variable, as having a constant term of c. If the constant term is 0, then it will conventionally be omitted when the quadratic is written out. Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x^0. In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial :x^2+2xy+y^2-2x+2y-4\ has a constant term of −4, which can be considered to be the coefficient of x^0y^0, where the va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless quantity, number without units, in which case it is known as a numerical factor. It may also be a constant (mathematics), constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any mathematical expression, expression (including Variable (mathematics), variables such as , and ). When the combination of variables and constants is not necessarily involved in a product (mathematics), product, it may be called a ''parameter''. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. A , also known as constant term or simply constant, is a quantity either implicitly attach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Rule (generalized Product Rule)
In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule for the derivative of the product of two (which is also known as "Leibniz's rule"). It states that if f and g are -times differentiable functions, then the product fg is also -times differentiable and its -th derivative is given by (fg)^=\sum_^n f^ g^, where = is the binomial coefficient and f^ denotes the ''j''th derivative of ''f'' (and in particular f^= f). The rule can be proven by using the product rule and mathematical induction. Second derivative If, for example, , the rule gives an expression for the second derivative of a product of two functions: (fg)''(x)=\sum\limits_^=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x). More than two factors The formula can be generalized to the product of ''m'' differentiable functions ''f''1,...,''f''''m''. \left(f_1 f_2 \cdots f_m\right)^=\sum_ \prod_f_^\,, where the sum extends over all ''m''-tuples (''k''1,...,''k''''m'') of non-negative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...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] |
