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11: 1.5 Calculus of Two or More Variables
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►A function is continuous on a point set
if it is continuous at all points of .
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►If is continuous, and is the set
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►Similarly, if is the set
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►If can be represented in both forms (1.5.30) and (1.5.33), and is continuous on , then
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►Infinite double integrals occur when becomes infinite at points in or when is unbounded.
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12: 27.21 Tables
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►Glaisher (1940) contains four tables: Table I tabulates, for all : (a) the canonical factorization of into powers of primes; (b) the Euler totient ; (c) the divisor function ; (d) the sum
of these divisors.
…Table III lists all solutions of the equation , and Table IV lists all solutions of the equation for all .
…6 lists , and for ; Table 24.
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13: 12.1 Special Notation
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►An older notation, due to Whittaker (1902), for is .
The notations are related by .
Whittaker’s notation is useful when is a nonnegative integer (Hermite polynomial case).
14: 21.6 Products
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21.6.2
►that is, is the number of elements in the set containing all -dimensional vectors obtained by multiplying on the right by a vector with integer elements.
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21.6.3
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21.6.4
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21.6.6
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15: 24.10 Arithmetic Properties
16: 19.29 Reduction of General Elliptic Integrals
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►where the arguments of the function are, in order, , , .
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►The only cases that are integrals of the third kind are those in which at least one with is a negative integer and those in which and is a positive integer.
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►where
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17: 19.20 Special Cases
18: 24.19 Methods of Computation
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Tanner and Wagstaff (1987) derives a congruence for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).
19: 27.2 Functions
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►It can be expressed as a sum over all primes :
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►the sum of the th powers of the positive integers that are relatively prime to .
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Assume that is dense in , i.
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, corresponding to distinct eigenvalues, are orthogonal: i.
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►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that , as and satisfy the same boundary conditions and thus define the same domains.
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►, and for .
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► We have a direct sum of linear spaces: .
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