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31—40 of 69 matching pages
31: 7.21 Physical Applications
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►Voigt functions , , can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects.
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32: 12.11 Zeros
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►If , then has no positive real zeros, and if , , then has a zero at .
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►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii).
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33: 28.33 Physical Applications
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►with , reduces to (28.32.2) with .
…The separated solutions must be -periodic in , and have the form
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28.33.2
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34: 18.39 Applications in the Physical Sciences
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►The properties of determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18.
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►where is assumed to be independent of time.
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►Now use spherical coordinates (1.5.16) with instead of , and assume the potential to be radial.
Then write instead of .
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►Analogous to (18.39.8) the 3D time-independent Schrödinger equation with potential is
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35: 10.17 Asymptotic Expansions for Large Argument
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10.17.14
►where denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that changes monotonically.
Bounds for are given by
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10.17.15
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►The bounds (10.17.15) also apply to in the conjugate sectors.
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36: 18.9 Recurrence Relations and Derivatives
37: 10.40 Asymptotic Expansions for Large Argument
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10.40.11
►where denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that changes monotonically.
Bounds for are given by
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10.40.12
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38: 18.3 Definitions
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39: 1.6 Vectors and Vector-Valued Functions
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►Suppose is a piecewise smooth surface which forms the complete boundary of a bounded closed point set , and is oriented by its normal being outwards from .
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1.6.58
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1.6.59
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1.6.60
►where is the derivative of normal to the surface outwards from and is the unit outer normal vector.
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