symmetric%0Aelliptic%20integrals
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21: 19.17 Graphics
§19.17 Graphics
►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. … ►For , , and , which are symmetric in , we may further assume that is the largest of if the variables are real, then choose , and consider only and . The cases or correspond to the complete integrals. … ►To view and for complex , put , use (19.25.1), and see Figures 19.3.7–19.3.12. …22: 19.20 Special Cases
§19.20 Special Cases
… ►The general lemniscatic case is … ►where may be permuted. … ►The general lemniscatic case is … ►23: 19.25 Relations to Other Functions
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§19.25(i) Legendre’s Integrals as Symmetric Integrals
… ►§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
… ► … ►§19.25(vii) Hypergeometric Function
… ►24: 19.28 Integrals of Elliptic Integrals
§19.28 Integrals of Elliptic Integrals
►In (19.28.1)–(19.28.3) we assume . … ►
19.28.6
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►To replace a single component of in by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)).
25: 19.34 Mutual Inductance of Coaxial Circles
§19.34 Mutual Inductance of Coaxial Circles
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19.34.3
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19.34.5
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19.34.6
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►References for other inductance problems solvable in terms of elliptic integrals are given in Grover (1946, pp. 8 and 283).
26: 19.37 Tables
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►Tabulated for to 6D by Byrd and Friedman (1971), to 15D for and 9D for by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964).
►Tabulated for to 10D by Fettis and Caslin (1964), and for to 7D by Zhang and Jin (1996, p. 673).
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►( is presented as .)
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§19.37(iv) Symmetric Integrals
… ►For check values of symmetric integrals with real or complex variables to 14S see Carlson (1995).27: 35.5 Bessel Functions of Matrix Argument
28: 19.32 Conformal Map onto a Rectangle
§19.32 Conformal Map onto a Rectangle
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19.32.1
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29: 19.1 Special Notation
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►All derivatives are denoted by differentials, not by primes.
►The first set of main functions treated in this chapter are Legendre’s complete integrals
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►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF.
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, , and are the symmetric (in , , and ) integrals of the first, second, and third kinds; they are complete if exactly one of , , and is identically 0.
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►The first three functions are incomplete integrals of the first, second, and third kinds, and the function includes complete integrals of all three kinds.
30: 35.1 Special Notation
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►Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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space of all real symmetric matrices. | |
real symmetric matrices. | |
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space of positive-definite real symmetric matrices. | |
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complex symmetric matrix. | |
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