complex%20variables
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11—20 of 34 matching pages
11: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
►When , has a string of complex zeros that approaches the ray as , and a conjugate string. … ►
12.11.1
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12.11.8
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12.11.9
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12: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Polynomial approximations in the complex plane.
J. Comput. Appl. Math. 18 (2), pp. 193–211.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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An Introduction to the Theory of Functions of a Complex Variable.
Oxford University Press, Oxford.
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13: 25.5 Integral Representations
14: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I.
Technical report
Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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15: 28.35 Tables
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§28.35(i) Real Variables
… ►Ince (1932) includes eigenvalues , , and Fourier coefficients for or , ; 7D. Also , for , , corresponding to the eigenvalues in the tables; 5D. Notation: , .
Kirkpatrick (1960) contains tables of the modified functions , for , , ; 4D or 5D.
National Bureau of Standards (1967) includes the eigenvalues , for with , and with ; Fourier coefficients for and for , , respectively, and various values of in the interval ; joining factors , for with (but in a different notation). Also, eigenvalues for large values of . Precision is generally 8D.
§28.35(ii) Complex Variables
…16: 9.7 Asymptotic Expansions
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►Numerical values of are given in Table 9.7.1 for to 2D.
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§9.7(iii) Error Bounds for Real Variables
… ►In (9.7.7) and (9.7.8) the th error term is bounded in magnitude by the first neglected term multiplied by where for (9.7.7) and for (9.7.8), provided that in the first case and in the second case. … ►§9.7(iv) Error Bounds for Complex Variables
… ►provided that , for (9.7.5) and , for (9.7.6). …17: 5.11 Asymptotic Expansions
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►Wrench (1968) gives exact values of up to .
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►where and are both fixed, and
…where is fixed, and is the Bernoulli polynomial defined in §24.2(i).
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►If is complex, then the remainder terms are bounded in magnitude by for (5.11.1), and for (5.11.2), times the first neglected terms.
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►In this subsection , , and are real or complex constants.
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18: 23.9 Laurent and Other Power Series
19: 18.40 Methods of Computation
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18.40.4
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18.40.5
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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18.40.7
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18.40.9
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20: 20.11 Generalizations and Analogs
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►where and .
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20.11.4
►In the case identities for theta functions become identities in the complex variable
, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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►However, in this case is no longer regarded as an independent complex variable within the unit circle, because is related to the variable
of the theta functions via (20.9.2).
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20.11.6
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