removable discontinuity
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31—40 of 52 matching pages
31: 25.2 Definition and Expansions
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25.2.4
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32: Errata
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Subsection 14.3(iv)
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Equation (5.11.14)
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Section 36.1 Special Notation
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Equation (25.2.4)
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References
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A sentence was added at the end of this subsection indicating that from (15.9.15), it follows that and are removable singularities.
The previous constraint was removed, see Fields (1966, (3)).
The entry for to represent complex conjugation was removed (see Version 1.0.19).
The original constraint, , was removed because, as stated after (25.2.1), is meromorphic with a simple pole at , and therefore is an entire function.
Suggested by John Harper.
33: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where is continuous, with convergence to if is an isolated point of discontinuity.
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►This is the discontinuity across the branch cut in (1.18.52) , from below to above the cut, divided by .
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►This representation has poles with residues at the discrete eigenvalues and a branch cut along with discontinuity, from below to above the cut, , as in (1.18.53), see Newton (2002, §7.1.1).
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34: 14.3 Definitions and Hypergeometric Representations
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►From (15.9.15) it follows that and are removable singularities of the right-hand sides of (14.3.21) and (14.3.22).
35: 19.29 Reduction of General Elliptic Integrals
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►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as multiplied either by or by in the cases of (19.29.20) or (19.29.21), respectively.
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36: 18.3 Definitions
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37: 19.21 Connection Formulas
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19.21.10
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38: 21.7 Riemann Surfaces
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►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann
surface. All compact Riemann surfaces can be obtained this
way.
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39: 33.14 Definitions and Basic Properties
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40: Bibliography B
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Uniform asymptotic smoothing of Stokes’s discontinuities.
Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
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