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1: 6.1 Special Notation
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►The main functions treated in this chapter are the exponential integrals , , and ; the logarithmic integral ; the sine integrals and ; the cosine integrals and .
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2: 8.21 Generalized Sine and Cosine Integrals
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►From §§8.2(i) and 8.2(ii) it follows that each of the four functions , , , and is a multivalued function of with branch point at .
Furthermore, and are entire functions of , and and are meromorphic functions of with simple poles at and , respectively.
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►When (and when , in the case of , or , in the case of ) the principal values of , , , and are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)).
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►From here on it is assumed that unless indicated otherwise the functions , , , and have their principal values.
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►For the corresponding expansions for and apply (8.21.20) and (8.21.21).
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3: 6.15 Sums
4: 6.21 Software
5: 6.14 Integrals
6: 6.19 Tables
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Abramowitz and Stegun (1964, Chapter 5) includes , , , , ; , , , , ; , , , , ; , , , , ; , , . Accuracy varies but is within the range 8S–11S.
Zhang and Jin (1996, pp. 652, 689) includes , , , 8D; , , , 8S.
7: 6.3 Graphics
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8: 6.13 Zeros
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and each have an infinite number of positive real zeros, which are denoted by , , respectively, arranged in ascending order of absolute value for .
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6.13.2
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9: 6.2 Definitions and Interrelations
10: 8.1 Special Notation
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►The functions treated in this chapter are the incomplete gamma functions , , , , and ; the incomplete beta functions and ; the generalized exponential integral ; the generalized sine and cosine integrals , , , and .
►Alternative notations include: Prym’s functions
, , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); , , Dingle (1973); , , Magnus et al. (1966); , , Luke (1975).