isolated essential singularity
(0.000 seconds)
11—20 of 98 matching pages
11: 31.12 Confluent Forms of Heun’s Equation
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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
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►This has regular singularities at and , and an irregular singularity of rank 1 at .
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►This has irregular singularities at and , each of rank .
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►This has a regular singularity at , and an irregular singularity at of rank .
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►This has one singularity, an irregular singularity of rank at .
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12: 31.14 General Fuchsian Equation
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►The general second-order Fuchsian equation with regular singularities at , , and at , is given by
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31.14.1
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►The exponents at the finite singularities
are and those at are , where
…The three sets of parameters comprise the singularity parameters
, the exponent parameters
, and the free accessory parameters
.
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31.14.3
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13: 34.12 Physical Applications
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►The angular momentum coupling coefficients (, , and symbols) are essential in the fields of nuclear, atomic, and molecular physics.
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14: 31.1 Special Notation
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►Sometimes the parameters are suppressed.
15: 16.21 Differential Equation
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►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at and an irregular singularity at .
When the only singularities of (16.21.1) are regular singularities at , , and .
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16: Notices
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►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education.
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17: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
… ►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities . …18: 36.14 Other Physical Applications
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►These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge.
Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns.
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19: 12.16 Mathematical Applications
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►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi).
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