properties of solutions

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11: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. Some properties are included as special cases of properties given in §31.15 below.

12: Mark J. Ablowitz

… ►Their similarity solutions lead to special ODEs which have the Painlevé property; i. …ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …

13: 28.2 Definitions and Basic Properties

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§28.2(ii) Basic Solutions $w_{\text{I}}$, $w_{\text{II}}$

… ►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. …

14: 28.29 Definitions and Basic Properties

… ►Then (28.29.1) has a nontrivial solution $w(z)$ with the pseudoperiodic property …

15: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

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§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

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§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

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16: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

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§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

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17: 29.6 Fourier Series

… ►In the special case $\nu =2n$, $m=0,1,\mathrm{\dots },n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\mathrm{\dots }$. …

18: 2.7 Differential Equations

… ►The solutions with the properties (2.7.26), (2.7.27) are unique, but not those with the properties (2.7.28), (2.7.29). …

19: 3.6 Linear Difference Equations

… ►The normalizing factor $\mathrm{\Lambda }$ can be the true value of $w_0$ divided by its trial value, or $\mathrm{\Lambda }$ can be chosen to satisfy a known property of the wanted solution of the form …

20: 32.2 Differential Equations

… ►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …