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1: 7.23 Tables
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Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Finn and Mugglestone (1965) includes the Voigt function , , , 6S.
Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
2: 29.9 Stability
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►If is not an integer, then (29.2.1) is unstable iff or lies in one of the closed intervals with endpoints and , .
If is a nonnegative integer, then (29.2.1) is unstable iff or for some .
3: 18.32 OP’s with Respect to Freud Weights
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►All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with , where the term half-Freud weight is used; or on or , where the term Rys weight is employed, see Rys et al. (1983).
For (generalized) Freud weights on a subinterval of see also Levin and Lubinsky (2005).
4: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
… ►Jacobian elliptic functions with real moduli in the intervals and , or with purely imaginary moduli are related to functions with moduli in the interval by the following formulas. … ►For proofs of these results and further information see Walker (2003).5: Mathematical Introduction
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complex plane (excluding infinity). | |
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is continuous at all points of a simple closed contour in . | |
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closed interval in , or closed straight-line segment joining and in . |
or | half-closed intervals. |
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6: About Color Map
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►Mathematically, we scale the height to lying in the interval and the components are computed as follows
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►Specifically, by scaling the phase angle in to in the interval , the hue (in degrees) is computed as
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7: 4.37 Inverse Hyperbolic Functions
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4.37.17
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4.37.18
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►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on .
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4.37.22
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4.37.24
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8: 5.14 Multidimensional Integrals
9: 22.18 Mathematical Applications
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►With the mapping gives a conformal map of the closed rectangle onto the half-plane , with mapping to respectively.
The half-open rectangle maps onto cut along the intervals and .
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10: 7.20 Mathematical Applications
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