completely multiplicative
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1: 27.3 Multiplicative Properties
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27.3.8
►A function is completely multiplicative if and
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►If is completely multiplicative, then (27.3.2) becomes
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2: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►For a completely multiplicative function we use the values at the primes together with (27.3.10).
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3: 27.4 Euler Products and Dirichlet Series
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►If is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes
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►The completely multiplicative function gives the Euler product representation of the Riemann zeta function
(§25.2(i)):
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4: 27.8 Dirichlet Characters
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►If
is a given integer, then a function is called a Dirichlet character (mod ) if it is completely multiplicative, periodic with period , and vanishes when .
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5: 22.8 Addition Theorems
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►If sums/differences of the ’s are rational multiples of , then further relations follow.
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6: 22.5 Special Values
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►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its -derivative (or at a pole, the residue), for values of that are integer multiples of , .
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7: 3.2 Linear Algebra
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►To an eigenvalue of multiplicity
, there correspond linearly independent eigenvectors provided that is nondefective, that is, has a complete set of linearly independent eigenvectors.
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8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Such orthonormal sets are called complete.
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►, of unit multiplicity, unless otherwise stated.
…and completeness implies
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►and completeness relation
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9: 23.1 Special Notation
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►The main functions treated in this chapter are the Weierstrass -function ; the Weierstrass zeta function ; the Weierstrass sigma function ; the elliptic modular function ; Klein’s complete invariant ; Dedekind’s eta function .
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lattice in . | |
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, | complete elliptic integrals (§19.2(i)). |
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set of all integer multiples of . | |
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10: Errata
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Equation (17.6.1)
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Equation (17.11.2)
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Equation (17.4.6)
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Equations (17.2.22) and (17.2.23)
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References
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17.6.1
The constraint was added.
17.11.2
The factor originally used in the denominator has been corrected to be .
The multi-product notation in the denominator of the right-hand side was used.
17.2.22
17.2.23
The numerators of the leftmost fractions were corrected to read and instead of and , respectively.
Reported 2017-06-26 by Jason Zhao.
An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).