machine epsilon
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1: 22.16 Related Functions
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§22.16(ii) Jacobi’s Epsilon Function
►Integral Representations
… ►See Figure 22.16.2. … ►Quasi-Addition and Quasi-Periodic Formulas
… ►Relation to Theta Functions
…2: 3.1 Arithmetics and Error Measures
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►The machine epsilon
, that is, the distance between and the next larger machine number with is given by .
The machine precision is .
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►Symmetric rounding or rounding to nearest of gives or , whichever is nearer to , with maximum relative error equal to the machine precision .
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3: 27.15 Chinese Remainder Theorem
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►Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory.
Details of a machine program describing the method together with typical numerical results can be found in Newman (1967).
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4: 33.17 Recurrence Relations and Derivatives
5: 33.15 Graphics
§33.15 Graphics
►§33.15(i) Line Graphs of the Coulomb Functions and
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§33.15(ii) Surfaces of the Coulomb Functions , , , and
… ►6: 33.14 Definitions and Basic Properties
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►This includes , hence can be expanded in a convergent power series in in a neighborhood of (§33.20(ii)).
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§33.14(iv) Solutions and
… ►An alternative formula for is … ►§33.14(v) Wronskians
►With arguments suppressed, …7: 33.1 Special Notation
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►The main functions treated in this chapter are first the Coulomb radial functions , , (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions , , , (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.
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Curtis (1964a):
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Greene et al. (1979):
nonnegative integers. | |
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real parameters. | |
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, .
, , .