relation to modulus and phase
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11: 32.11 Asymptotic Approximations for Real Variables
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►where is the gamma function (§5.2(i)), and the branch of the function is immaterial.
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►where is an arbitrary constant such that , and
…The connection formulas relating (32.11.25) and (32.11.26) are
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►Now suppose .
…and the branch of the function is immaterial.
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12: 33.23 Methods of Computation
§33.23 Methods of Computation
… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►§33.23(iv) Recurrence Relations
►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer , provided that the recurrence is carried out in a stable direction (§3.6). … ► (12) should be ). …13: 25.10 Zeros
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►Calculations relating to the zeros on the critical line make use of the real-valued function
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25.10.2
►is chosen to make real, and assumes its principal value.
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►Sign changes of are determined by multiplying (25.9.3) by
to obtain the Riemann–Siegel formula:
…where as .
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14: 9.13 Generalized Airy Functions
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►Swanson and Headley (1967) define independent solutions and of (9.13.1) by
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►Their relations to the functions and are given by
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►The function on the right-hand side is recessive in the sector , and is therefore an essential member of any numerically satisfactory pair of solutions in this region.
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►When is a positive integer the relation of these functions to
, is as follows:
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►When is not an integer the branch of in (9.13.25) is usually chosen to be with .
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15: 15.9 Relations to Other Functions
§15.9 Relations to Other Functions
►§15.9(i) Orthogonal Polynomials
… ►Jacobi
… ►Legendre
… ►Meixner
…16: 36.7 Zeros
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►The zeros in Table 36.7.1 are points in the plane, where is undetermined.
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►The zeros are lines in space where is undetermined.
…The rings are almost circular (radii close to
and varying by less than 1%), and almost flat (deviating from the planes by at most ).
…Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral.
There are also three sets of zero lines in the plane
related by rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates is given by
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17: 8.19 Generalized Exponential Integral
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§8.19(i) Definition and Integral Representations
… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►with in both equations. … ►§8.19(v) Recurrence Relation and Derivatives
… ►§8.19(vi) Relation to Confluent Hypergeometric Function
…18: 9.11 Products
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►For example, , , , .
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►
9.11.4
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►For related integrals see Gordon (1969, Appendix B).
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►
9.11.15
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►
9.11.19
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19: 25.12 Polylogarithms
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►When , , (25.12.1) becomes
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►The special case is the Riemann zeta function: .
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►Further properties include
…and
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►In terms of polylogarithms
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