exponent%20pairs
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1: 36.5 Stokes Sets
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36.5.4
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36.5.7
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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2: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…where are the distinct prime factors of , each exponent
is positive, and is the number of distinct primes dividing .
…Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►is the sum of the th powers of the divisors of , where the exponent
can be real or complex.
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►In the following examples, are the exponents in the factorization of in (27.2.1).
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3: 28.29 Definitions and Basic Properties
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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent
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28.29.9
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►Given together with the condition (28.29.6), the solutions of (28.29.9) are the characteristic
exponents of (28.29.1).
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►If
is a solution of (28.29.9), then , comprise a fundamental pair of solutions of Hill’s equation.
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4: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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5: 15.11 Riemann’s Differential Equation
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►Here , , are the exponent pairs at the points , , , respectively.
Cases in which there are fewer than three singularities are included automatically by allowing the choice for exponent pairs.
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6: 28.36 Software
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§28.36(ii) Characteristic Exponents and Eigenvalues
…7: 14.2 Differential Equations
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§14.2(iii) Numerically Satisfactory Solutions
►Equation (14.2.2) has regular singularities at , , and , with exponent pairs , , and , respectively; compare §2.7(i). … ►Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval . When , or , and are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. … ►Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval . …8: 28.34 Methods of Computation
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§28.34(i) Characteristic Exponents
…9: 3.1 Arithmetics and Error Measures
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►A nonzero normalized binary floating-point machine number
is represented as
…where is equal to or , each , , is either or , is the most significant bit, () is the number of significant bits , is the least significant bit, is an integer called the exponent, is the significand, and is the fractional
part.
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3.1.2
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►Let with and .
For given values of , , and , the format width in bits
of a computer word is the total number of bits: the sign (one bit), the significant bits ( bits), and the bits allocated to the exponent (the remaining bits).
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10: 15.10 Hypergeometric Differential Equation
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►It has regular singularities at , with corresponding exponent pairs
, , , respectively.
When none of the exponent pairs differ by an integer, that is, when none of , , is an integer, we have the following pairs
, of fundamental solutions.
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►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions.
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►The connection formulas for the principal branches of Kummer’s solutions are:
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