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Tanc Function
In mathematics, the tanc function is defined for z \neq 0 as \operatorname(z)=\frac Properties The first-order derivative of the tanc function is given by : \frac - \frac The Taylor series expansion is\operatorname z \approx \left(1+ \frac z^2 + \frac z^4 + \frac z^6 + \frac z^8 + \frac z^ + \frac z^+ \frac z^ + O(z^ ) \right)which leads to the series expansion of the integral as\int _0^z \frac \, dx = \left(z+ \frac z^3 + \frac z^5 + \frac z^7 + \frac z^9+ \frac z^+ \frac z^ + \frac z^+ O (z^) \right)The Padé approximant is\operatorname \left( z \right) = \left( 1-\,^ + \,^-\,^+\,^ \right) \left( 1-\,^+\,^-\,^+\,^ \right) ^ In terms of other special functions * \operatorname(z)=, where (a,b,z) is Kummer's confluent hypergeometric function. *\operatorname(z)= \frac , where (q, \alpha, \gamma, \delta, \epsilon ,z) is the biconfluent Heun function. * \operatorname(z)= \frac , where (a,b,z) is a Whittaker function. Gallery {, , , , See also * Sinhc f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods— in some sense inspired by the Padé theory— typically replace them. Since Padé approximant is a rational function, an artificial singul ... [...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] |
Heun Function
In mathematics, the local Heun function H \ell (a,q;\alpha ,\beta, \gamma, \delta ; z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point ''z'' = 0. The local Heun function is called a Heun function, denoted ''Hf'', if it is also regular at ''z'' = 1, and is called a Heun polynomial, denoted ''Hp'', if it is regular at all three finite singular points ''z'' = 0, 1, ''a''. Heun's equation Heun's equation is a second-order linear ordinary differential equation (ODE) of the form :\frac + \left frac+ \frac + \frac \right \frac + \frac w = 0. The condition \epsilon=\alpha+\beta-\gamma-\delta+1 is taken so that the characteristic exponents for the regular singularity at infinity are α and β (see below). The complex number ''q'' is called the accessory parameter. Heun's equation has four regular singular points: 0, 1, ''a'' and ∞ with exponents (0, 1 −&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whittaker Function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R). Whittaker's equation is :\frac+\left(-\frac+\frac+\frac\right)w=0. It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions ''M''κ,μ(''z''), ''W''κ,μ(''z''), defined in terms of Kummer's confluent hypergeometric functions ''M'' and ''U'' by :M_\left(z\right) = \exp\left(-z/2\right)z^M\left(\mu-\kappa+\tfrac, 1+2\mu, z\right) :W_\left(z\right) = \exp\left(-z/2\right)z^U\left(\mu-\kappa+\tfrac, 1+2\mu, z\right). The Whittaker function W_(z) is the same as those with opposite val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
