solution of a rational Schr�dinger equation

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1: 15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

… ►When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_1(z)$, $f_2(z)$ of fundamental solutions. … ► ►

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …

2: 31.2 Differential Equations

… ► $w(z)=z^{1-\gamma }{w}_1(z)$ satisfies (31.2.1) if $w_1$ is a solution of (31.2.1) with transformed parameters $q_1=q+(a\delta +ϵ)(1-\gamma )$; ${\alpha }_1=\alpha +1-\gamma$, ${\beta }_1=\beta +1-\gamma$, ${\gamma }_1=2-\gamma$. Next, $w(z)={(z-1)}^{1-\delta }{w}_2(z)$ satisfies (31.2.1) if $w_2$ is a solution of (31.2.1) with transformed parameters $q_2=q+a\gamma (1-\delta )$; ${\alpha }_2=\alpha +1-\delta$, ${\beta }_2=\beta +1-\delta$, ${\delta }_2=2-\delta$. Lastly, $w(z)={(z-a)}^{1-ϵ}{w}_3(z)$ satisfies (31.2.1) if $w_3$ is a solution of (31.2.1) with transformed parameters $q_3=q+\gamma (1-ϵ)$; ${\alpha }_3=\alpha +1-ϵ$, ${\beta }_3=\beta +1-ϵ$, ${ϵ}_3=2-ϵ$. … ►If $\tilde{z}=\tilde{z}(z)$ is one of the $3!=6$ homographies that map $\mathrm{\infty }$ to $\mathrm{\infty }$, then $w(z)=\tilde{w}(\tilde{z})$ satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and appropriately transformed parameters. …For example, $w(z)={(1-z)}^{-\alpha }\tilde{w}(z/(z-1))$, which arises from $\tilde{z}=z/(z-1)$, satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and transformed parameters $\tilde{a}=a/(a-1)$, $\tilde{q}=-(q-a\alpha \gamma )/(a-1)$; $\tilde{\beta }=\alpha +1-\delta$, $\tilde{\delta }=\alpha +1-\beta$. …

3: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

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4: 29.2 Differential Equations

§29.2 Differential Equations

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§29.2(i) Lamé’s Equation

… ►In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). … ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1).

5: 32.2 Differential Equations

… ►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions. … ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. …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. … ►Let …

6: 28.2 Definitions and Basic Properties

… ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with … ►

§28.2(iv) Floquet Solutions

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

7: 28.20 Definitions and Basic Properties

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§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

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§28.20(iii) Solutions ${\mathrm{M}}_{\nu }^{(j)}$

… ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions ${\mathrm{M}}_{\nu }^{(3)}(z,h)$, ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ such that … ►

§28.20(v) Solutions ${\mathrm{Ie}}_n$, ${\mathrm{Io}}_n$, ${\mathrm{Ke}}_n$, ${\mathrm{Ko}}_n$

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8: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

9: 32.8 Rational Solutions

§32.8 Rational Solutions

… ►Special rational solutions of ${\text{P}}_{\text{III}}$ are … ►Then ${\text{P}}_{\text{III}}$ has rational solutions iff … ►These rational solutions have the form … ►

10: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

… ►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).