solutions analytic at two singularities
(0.002 seconds)
1—10 of 16 matching pages
1: 31.1 Special Notation
…
►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
…
►This equation has a regular singularity at the origin with indices and , and an irregular singularity at infinity of rank one.
…In effect, the regular singularities of the hypergeometric differential equation at
and coalesce into an irregular singularity at
.
…
►The first two standard solutions are:
…
►
…
►
§13.2(ii) Analytic Continuation
…4: 33.2 Definitions and Basic Properties
…
►
§33.2(i) Coulomb Wave Equation
… ►This differential equation has a regular singularity at with indices and , and an irregular singularity of rank 1 at (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which (§2.8(i)). … ►§33.2(ii) Regular Solution
… ►As in the case of , the solutions and are analytic functions of when . …5: 4.13 Lambert -Function
…
►The Lambert -function is the solution of the equation
…
►On the -interval there are two real solutions, one increasing and the other decreasing.
…
►Other solutions of (4.13.1) are other branches of .
… is a single-valued analytic function on , real-valued when , and has a square root branch point at
.
…The other branches are single-valued analytic functions on , have a logarithmic branch point at
, and, in the case , have a square root branch point at
respectively.
…
6: 31.11 Expansions in Series of Hypergeometric Functions
…
►The Fuchs-Frobenius solutions at
are
…
►and (31.11.1) converges to (31.3.10) outside the ellipse in the -plane with foci at 0, 1, and passing through the third finite singularity at
.
►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
and has exponent
at
.
The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities
and .
…
7: 20.13 Physical Applications
…
►is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at
.
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 , the functions , , 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: 10.25 Definitions
…
►
►
§10.25(ii) Standard Solutions
… ►In particular, the principal branch of is defined in a similar way: it corresponds to the principal value of , is analytic in , and two-valued and discontinuous on the cut . … ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in , and two-valued and discontinuous on the cut . … ►§10.25(iii) Numerically Satisfactory Pairs of Solutions
…9: 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.
…
10: 16.8 Differential Equations
…
►is a value of
at which all the coefficients , , are analytic.
If is not an ordinary point but , , are analytic at
, then is a regular singularity.
…
►Equation (16.8.4) has a regular singularity at
, and an irregular singularity at
, whereas (16.8.5) has regular singularities at
, , and .
…
►When no is an integer, and no two
differ by an integer, a fundamental set of solutions of (16.8.3) is given by
…
►Thus in the case the regular singularities of the function on the left-hand side at
and coalesce into an irregular singularity at
.
…