regularization
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1: 33.8 Continued Fractions
2: 33.3 Graphics
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§33.3(i) Line Graphs of the Coulomb Radial Functions and
► ► ► … ►§33.3(ii) Surfaces of the Coulomb Radial Functions and
…3: 25.17 Physical Applications
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►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect).
It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).
4: 33.23 Methods of Computation
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►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii and , respectively, and may be used to compute the regular and irregular solutions.
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►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii.
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►This implies decreasing for the regular solutions and increasing for the irregular solutions of §§33.2(iii) and 33.14(iii).
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►§33.8 supplies continued fractions for and .
Combined with the Wronskians (33.2.12), the values of , , and their derivatives can be extracted.
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5: 33.2 Definitions and Basic Properties
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§33.2(i) Coulomb Wave Equation
… ►This differential equation has a regular singularity at with indices and , and an irregular singularity of rank 1 at (§§2.7(i), 2.7(ii)). … ►§33.2(ii) Regular Solution
►The function is recessive (§2.7(iii)) at , and is defined by … ► is a real and analytic function of on the open interval , and also an analytic function of when . …6: 33.5 Limiting Forms for Small , Small , or Large
7: 33.24 Tables
8: 33.10 Limiting Forms for Large or Large
9: 33.16 Connection Formulas
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§33.16(i) and in Terms of and
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33.16.1
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§33.16(ii) and in Terms of and when
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33.16.4
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