# solutions analytic at two singularities

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##### 1: 31.1 Special Notation
Sometimes the parameters are suppressed.
##### 3: 13.2 Definitions and Basic Properties
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 $\infty$ coalesce into an irregular singularity at $\infty$. … The first two standard solutions are: …
##### 4: 33.2 Definitions and Basic Properties
###### §33.2(i) Coulomb Wave Equation
This differential equation has a regular singularity at $\rho=0$ with indices $\ell+1$ and $-\ell$, and an irregular singularity of rank 1 at $\rho=\infty$ (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which $\ifrac{{\mathrm{d}}^{2}w}{{\mathrm{d}\rho}^{2}}=0$2.8(i)). …
###### §33.2(ii) Regular Solution$F_{\ell}\left(\eta,\rho\right)$
As in the case of $F_{\ell}\left(\eta,\rho\right)$, the solutions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ are analytic functions of $\rho$ when $0<\rho<\infty$. …
##### 5: 31.11 Expansions in Series of Hypergeometric Functions
Then the Fuchs–Frobenius solution at $\infty$ 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 $\infty$. 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 $z=1$. …
##### 6: 4.13 Lambert $W$-Function
The Lambert $W$-function $W\left(x\right)$ is the solution of the equation … On the $x$-interval $[0,\infty)$ there is one real solution, and it is nonnegative and increasing. On the $x$-interval $(-1/e,0)$ there are two real solutions, one increasing and the other decreasing. We call the solution for which $W\left(x\right)\geq W\left(-1/e\right)$ 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
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 $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, 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
The equation … Assume that in the equation … 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
###### §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 $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
##### 10: 16.8 Differential Equations
is a value $z_{0}$ of $z$ at which all the coefficients $f_{j}(z)$, $j=0,1,\dots,n-1$, are analytic. If $z_{0}$ is not an ordinary point but $(z-z_{0})^{n-j}f_{j}(z)$, $j=0,1,\dots,n-1$, 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 $z=\infty$, whereas (16.8.5) has regular singularities at $z=0$, $1$, and $\infty$. … 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 $\infty$ coalesce into an irregular singularity at $\infty$. …