# solutions analytic at two singularities

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##### 1: 31.1 Special Notation

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►Sometimes the parameters are suppressed.

##### 2: 31.4 Solutions Analytic at Two Singularities: Heun Functions

###### §31.4 Solutions Analytic at Two Singularities: Heun Functions

…##### 3: 13.2 Definitions and Basic Properties

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►This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one.
…In effect, the regular singularities of the hypergeometric differential equation at
$b$ and $\mathrm{\infty}$ coalesce into an irregular singularity at
$\mathrm{\infty}$.
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►The first two standard solutions are:
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###### §13.2(ii) Analytic Continuation

…##### 4: 33.2 Definitions and Basic Properties

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###### §33.2(i) Coulomb Wave Equation

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell}+1$ and $-\mathrm{\ell}$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty}$ (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which ${d}^{2}w/{d\rho}^{2}=0$ (§2.8(i)). … ►###### §33.2(ii) Regular Solution ${F}_{\mathrm{\ell}}(\eta ,\rho )$

… ►As in the case of ${F}_{\mathrm{\ell}}(\eta ,\rho )$, the solutions ${H}_{\mathrm{\ell}}^{\pm}(\eta ,\rho )$ and ${G}_{\mathrm{\ell}}(\eta ,\rho )$ are analytic functions of $\rho $ when $$. …##### 5: 31.11 Expansions in Series of Hypergeometric Functions

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►Then the Fuchs–Frobenius solution at
$\mathrm{\infty}$ belonging to the exponent $\alpha $ has the expansion (31.11.1) with
…and (31.11.1) converges outside the ellipse $\mathcal{E}$ in the $z$-plane with foci at 0, 1, and passing through the third finite singularity at
$z=a$.
►Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function.
►For example, consider the Heun function which is analytic at
$z=a$ and has exponent $\alpha $
at
$\mathrm{\infty}$.
The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities
$z=0$ and $z=1$.
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##### 6: 4.13 Lambert $W$-Function

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►The Lambert $W$-function $W\left(x\right)$ is the solution of the equation
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►On the $x$-interval $[0,\mathrm{\infty})$ there is one real solution, and it is nonnegative and increasing.
On the $x$-interval $(-1/\mathrm{e},0)$ there are two real solutions, one increasing and the other decreasing.
We call the solution for which $W\left(x\right)\ge W\left(-1/\mathrm{e}\right)$ the

*principal branch*and denote it by $\mathrm{Wp}\left(x\right)$. The other solution is denoted by $\mathrm{Wm}\left(x\right)$. …##### 7: 20.13 Physical Applications

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►is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at
$t=0$.
These two apparently different solutions differ only in their normalization and boundary conditions.
…Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281).
►In the singular limit $\mathrm{\Im}\tau \to 0+$, the functions ${\theta}_{j}\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195).
This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

##### 8: 1.13 Differential Equations

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►The equation
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►Assume that in the equation
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►The

*inhomogeneous*(or*nonhomogeneous*) equation … ►The product of any two solutions of (1.13.1) satisfies … ►For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …##### 9: 10.25 Definitions

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###### §10.25(ii) Standard Solutions

… ►In particular, the*principal branch*of ${I}_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu}$, is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►The*principal branch*corresponds to the principal value of the square root in (10.25.3), is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►###### §10.25(iii) Numerically Satisfactory Pairs of Solutions

…##### 10: 16.8 Differential Equations

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►is a value ${z}_{0}$ of $z$
at which all the coefficients ${f}_{j}(z)$, $j=0,1,\mathrm{\dots},n-1$, are analytic.
If ${z}_{0}$ is not an ordinary point but ${(z-{z}_{0})}^{n-j}{f}_{j}(z)$, $j=0,1,\mathrm{\dots},n-1$, are analytic at
$z={z}_{0}$, then ${z}_{0}$ is a

*regular singularity*. … ►Equation (16.8.4) has a regular singularity at $z=0$, and an irregular singularity at $z=\mathrm{\infty}$, whereas (16.8.5) has regular singularities at $z=0$, $1$, and $\mathrm{\infty}$. … ►When no ${b}_{j}$ is an integer, and no two ${b}_{j}$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … ►Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha $ and $\mathrm{\infty}$ coalesce into an irregular singularity at $\mathrm{\infty}$. …