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11: 10.74 Methods of Computation
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►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument or is sufficiently small in absolute value.
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►Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection.
►For large positive real values of the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used.
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►In the interval either direction of integration can be used for both functions.
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►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter.
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12: 10.50 Wronskians and Cross-Products
13: 25.10 Zeros
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►Because , vanishes at the zeros of , which can be separated by observing sign changes of .
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►By comparing with the number of sign changes of we can decide whether has any zeros off the line in this region.
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►The error term
can be expressed as an asymptotic series that begins
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►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)).
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14: 1.16 Distributions
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►Let be a function defined on an open interval , which can be infinite.
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►We denote it by .
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►, a function on which is absolutely Lebesgue integrable on every compact subset of ) such that
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►The partial derivatives of distributions in
can be defined as in §1.16(ii).
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15: 2.4 Contour Integrals
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(b)
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►Higher coefficients in (2.4.15) can be found from (2.3.18) with , , and replaced by .
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►Consider the integral
…The problem of obtaining an asymptotic approximation to that is uniform with respect to in a region containing is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v).
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►For large , is approximated uniformly by the integral that corresponds to (2.4.19) when is replaced by a constant.
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ranges along a ray or over an annular sector , , where , , and . converges at absolutely and uniformly with respect to .
16: 2.11 Remainder Terms; Stokes Phenomenon
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►As an example consider
…But this answer is incorrect: to 7D .
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►Where should the change-over take place? Can it be accomplished smoothly?
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►Expansions similar to (2.11.15) can be constructed for many other special functions.
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►Often the process of re-expansion can be repeated any number of times.
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17: 10.38 Derivatives with Respect to Order
18: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function for .
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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function , and the values can be checked by the congruence (27.14.20).
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19: 10.57 Uniform Asymptotic Expansions for Large Order
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►Asymptotic expansions for , , , , , and as that are uniform with respect to
can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9).
Subsequently, for the connection formula (10.47.11) is available.
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20: 10.47 Definitions and Basic Properties
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►(This is in contrast to other treatments of spherical Bessel functions, including Abramowitz and Stegun (1964, Chapter 10), in which
can be any integer.
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, , and are the modified spherical Bessel functions.
►Many properties of , , , , , , and follow straightforwardly from the above definitions and results given in preceding sections of this chapter.
For example, , , , , , , and are all entire functions of .
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►For (10.47.2) numerically satisfactory pairs of solutions are and in the right half of the -plane, and and in the left half of the -plane.
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