point sets in complex plane
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21: 23.20 Mathematical Applications
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►The interior of is mapped one-to-one onto the lower half-plane.
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►It follows from the addition formula (23.10.1) that the points
, , have zero sum iff , so that addition of points on the curve corresponds to addition of parameters on the torus ; see McKean and Moll (1999, §§2.11, 2.14).
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►If , then intersects the plane
in a curve that is connected if ; if , then the intersection has two components, one of which is a closed loop.
…The addition law states that to find the sum of two points, take the third intersection with of the chord joining them (or the tangent if they coincide); then its reflection in the -axis gives the required sum.
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►Let denote the set of points on that are of finite order (that is, those points
for which there exists a positive integer with ), and let be the sets of points with integer and rational coordinates, respectively.
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22: 4.23 Inverse Trigonometric Functions
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►In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points
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The function assumes its principal value when ; elsewhere on the integration paths the branch is determined by continuity.
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►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the -plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts.
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►These functions are analytic in the cut plane depicted in Figures 4.23.1(iii) and 4.23.1(iv).
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►For example, from the heading and last entry in the penultimate column we have .
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23: 15.6 Integral Representations
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►In (15.6.2) the point
lies outside the integration contour, and assume their principal values where the contour cuts the interval , and at .
►In (15.6.3) the point
lies outside the integration contour, the contour cuts the real axis between and , at which point
and .
►In (15.6.4) the point
lies outside the integration contour, and at the point where the contour cuts the negative real axis and .
►In (15.6.5) the integration contour starts and terminates at a point
on the real axis between and .
…At the starting point
and are zero.
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24: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►With denoting here the elementary charge, the Coulomb potential between two point particles with charges and masses separated by a distance is , where are atomic numbers, is the electric constant, is the fine structure constant, and is the reduced Planck’s constant.
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►Resolution of the ambiguous signs in (33.22.11), (33.22.12) depends on the sign of
in (33.22.3).
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•
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§33.22(vi) Solutions Inside the Turning Point
… ►The Coulomb functions given in this chapter are most commonly evaluated for real values of , , , and nonnegative integer values of , but they may be continued analytically to complex arguments and order as indicated in §33.13. … ►Searches for resonances as poles of the -matrix in the complex half-plane . See for example Csótó and Hale (1997).
25: Bibliography D
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point.
Anal. Appl. (Singap.) 12 (4), pp. 385–402.
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26: 4.2 Definitions
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►This is a multivalued function of with branch point at .
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is a single-valued analytic function on and real-valued when ranges over the positive real numbers.
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►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by , the other set corresponding to the “lower side” and denoted by .
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►In all other cases, is a multivalued function with branch point at .
…This is an analytic function of on , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless .
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27: 4.13 Lambert -Function
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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.
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►in which the are polynomials of degree with
…where is defined in §5.11(i).
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►where .
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28: 22.18 Mathematical Applications
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§22.18(i) Lengths and Parametrization of Plane Curves
… ►With the mapping gives a conformal map of the closed rectangle onto the half-plane , with mapping to respectively. … ►in which are real constants, can be achieved in terms of single-valued functions. …Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). … ►For any two points and on this curve, their sum , always a third point on the curve, is defined by the Jacobi–Abel addition law …29: 14.21 Definitions and Basic Properties
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and exist for all values of , , and , except possibly and , which are branch points (or poles) of the functions, in general.
When is complex
, , and are defined by (14.3.6)–(14.3.10) with replaced by : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when , and by continuity elsewhere in the -plane with a cut along the interval ; compare §4.2(i).
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►When and , a numerically satisfactory pair of solutions of (14.21.1) in the half-plane
is given by and .
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►Many of the properties stated in preceding sections extend immediately from the -interval to the cut -plane
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…The generating function expansions (14.7.19) (with replaced by ) and (14.7.22) apply when ; (14.7.21) (with replaced by ) applies when .
30: 1.14 Integral Transforms
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►Suppose is a real- or complex-valued function and is a real or complex parameter.
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►Moreover, if
in some half-plane
and , then (1.14.20) holds for .
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►The Mellin transform of a real- or complex-valued function is defined by
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►If the integral converges, then it converges uniformly in any compact domain in the complex
-plane not containing any point of the interval .
In this case, represents an analytic function in the -plane cut along the negative real axis, and
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