Liouville transformation for differential equations
(0.003 seconds)
1—10 of 17 matching pages
1: 1.13 Differential Equations
Liouville Transformation
…2: 2.8 Differential Equations with a Parameter
3: Bibliography E
4: 2.7 Differential Equations
§2.7(iii) Liouville–Green (WKBJ) Approximation
►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: ►Liouville–Green Approximation Theorem
… ►The first of these references includes extensions to complex variables and reversions for zeros. …5: 2.6 Distributional Methods
§2.6(ii) Stieltjes Transform
… ►The Stieltjes transform of $f(t)$ is defined by … ►Corresponding results for the generalized Stieltjes transform … ►The Riemann–Liouville fractional integral of order $\mu $ is defined by … ►where $\mathcal{M}f\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …6: 18.38 Mathematical Applications
Differential Equations: Spectral Methods
… ►This process has been generalized to spectral methods for solving partial differential equations. … ►The basic ideas of Gaussian quadrature, and their extensions to nonclassical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►Radon Transform
…7: Bibliography O
8: Errata
The following additions were made in Chapter 1:

§1.2
New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

§1.3
The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
 §1.4

§1.8
In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

§1.10
New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

§1.13
New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: SturmLiouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

§1.14(i)
Another form of Parseval’s formula, (1.14.7_5).

§1.16
We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

§1.17
Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

§1.18
An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).