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11: 4.37 Inverse Hyperbolic Functions
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►In (4.37.2) the integration path may not pass through either of the points
, and the function assumes its principal value when .
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12: 10.25 Definitions
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►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 .
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10.25.3
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►It has a branch point at for all .
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 .
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13: 23.20 Mathematical Applications
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►Points
on the curve can be parametrized by , , where and : in this case we write .
The curve is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element as the point at infinity, the negative of by , and generally on the curve iff the points
, , are collinear.
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).
►In terms of the addition law can be expressed , ; otherwise , where
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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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14: 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.
… and have branch points at ; the other four functions have branch points at .
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►Care needs to be taken on the cuts, for example, if then .
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►where and in (4.23.34) and (4.23.35), and in (4.23.36).
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15: 1.10 Functions of a Complex Variable
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►If and , then one branch is , the other branch is , with in both cases.
Similarly if , then one branch is , the other branch is , with in both cases.
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►
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►Alternatively, take to be any point in and set
where the logarithms assume their principal values.
(Thus if is in the interval , then the logarithms are real.)
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16: 36.5 Stokes Sets
§36.5 Stokes Sets
►§36.5(i) Definitions
►Stokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or . …where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …17: 18.2 General Orthogonal Polynomials
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Orthogonality on Countable Sets
►Let be a finite set of distinct points on , or a countable infinite set of distinct points on , and , , be a set of positive constants. …when is a finite set of distinct points. … ►If the orthogonality interval is or , then the role of can be played by , the central-difference operator in the imaginary direction (§18.1(i)). … ►If then the interval is included in the support of , and outside the measure only has discrete mass points such that are the only possible limit points of the sequence , see Máté et al. (1991, Theorem 10). …18: 13.4 Integral Representations
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►The contour of integration starts and terminates at a point
on the real axis between and .
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13.4.12
, .
►At the point where the contour crosses the interval , and the function assume their principal values; compare §§15.1 and 15.2(i).
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►
13.4.13
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►At this point the fractional powers are determined by and .
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19: 36.7 Zeros
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►The zeros in Table 36.7.1 are points in the plane, where is undetermined.
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►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the -axis that is far from the origin, the zero contours form an array of rings close to the planes
…The rings are almost circular (radii close to and varying by less than 1%), and almost flat (deviating from the planes by at most ).
…Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral.
There are also three sets of zero lines in the plane related by rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates is given by
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