# Liouville theorem

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## 1—10 of 15 matching pages

##### 1: 1.9 Calculus of a Complex Variable

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###### Liouville’s Theorem

…##### 2: 2.7 Differential Equations

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►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
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###### Liouville–Green Approximation Theorem

…##### 3: 15.16 Products

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##### 4: Bibliography S

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Sturm oscillation and comparison theorems.
In Sturm-Liouville theory,
pp. 29–43.
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##### 5: 27.2 Functions

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###### §27.2(i) Definitions

… ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the*prime number theorem*. … ►If $\left(a,n\right)=1$, then the*Euler–Fermat theorem*states that … ►This is*Liouville’s function*. …##### 6: 18.39 Applications in the Physical Sciences

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►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with ${L}^{2}$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty}$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the

*Sturm oscillation theorem*. …##### 7: 3.8 Nonlinear Equations

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►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013).
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##### 8: Errata

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►We now include Markov’s Theorem.
In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem.
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►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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Paragraph Prime Number Theorem (in §27.12)
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Subsections 1.15(vi), 1.15(vii), 2.6(iii)
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The largest known prime, which is a Mersenne prime, was updated from ${2}^{43,112,609}-1$ (2009) to ${2}^{82,589,933}-1$ (2018).

A number of changes were made with regard to fractional integrals and derivatives.
In §1.15(vi) a reference to Miller and Ross (1993) was added,
the fractional integral operator of order $\alpha $ was more precisely identified as the
*Riemann-Liouville* fractional integral operator of order $\alpha $, and a paragraph was added below
(1.15.50) to generalize (1.15.47).
In §1.15(vii) the sentence defining the fractional derivative was clarified.
In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator
was made consistent with §1.15(vi).

##### 9: 18.36 Miscellaneous Polynomials

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►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, ${A}_{n}{A}_{n-1}{C}_{n}>0$ for $n\ge 1$ as per (18.2.9_5).
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►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the ${L}_{n}^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness.
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►The $y(x)={\widehat{L}}_{n}^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients:
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►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a

*rational*potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …