Liouville–Green approximation theorem

SearchAdvancedHelp

(0.001 seconds)

31—40 of 271 matching pages

31: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

32: 16.26 Approximations

§16.26 Approximations

►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

33: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

►For asymptotic approximations for the accessory parameter eigenvalues $q_m$, see Fedoryuk (1991) and Slavyanov (1996). ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

34: 3.7 Ordinary Differential Equations

… ►

§3.7(iv) Sturm–Liouville Eigenvalue Problems

… ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …

35: 33.1 Special Notation

… ►

$f^{(0)}(ϵ,\mathrm{\ell };r)=f(ϵ,\mathrm{\ell };r)$, $f(ϵ,\mathrm{\ell };r)=s(ϵ,\mathrm{\ell };r)$, $g(ϵ,\mathrm{\ell };r)=c(ϵ,\mathrm{\ell };r)$.

36: 2.6 Distributional Methods

§2.6 Distributional Methods

… ►

§2.6(ii) Stieltjes Transform

… ►Corresponding results for the generalized Stieltjes transform … ►The Riemann–Liouville fractional integral of order $\mu$ is defined by … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

37: 1.15 Summability Methods

… ►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►

§1.15(viii) Tauberian Theorems

…

38: 10.21 Zeros

… ►The approximations that follow in §10.21(viii) do not suffer from this drawback. ►

§10.21(viii) Uniform Asymptotic Approximations for Large Order

►As $\nu \to \mathrm{\infty }$ the following four approximations hold uniformly for $m=1,2,\mathrm{\dots }$: … ►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013). … ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …

39: 10.76 Approximations

§10.76 Approximations

… ►

§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions

… ►

Bickley Functions

… ►

Spherical Bessel Functions

… ►

Kelvin Functions

…

40: About Color Map

… ►The following figure illustrates the piece-wise linear mapping of the height to each of the color components red, green and blue, written as $⟨R,G,B⟩$. … ►The four color scheme quickly indicates in which quadrant $z$ lies: the colors blue, green, red and yellow are used to indicate the first, second, third and fourth quadrants, respectively. … ►The conventional CMYK color wheel (not to be confused with the traditional Artist’s color wheel) places the additive colors (red, green, blue) and the subtractive colors (yellow, cyan, magenta) at multiples of 60 degrees. …