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1: 22.15 Inverse Functions
§22.15(ii) Representations as Elliptic Integrals
2: 19.26 Addition Theorems
Equivalent forms of (19.26.2) are given by …Equivalent forms of (19.26.11) are given by … Equivalent forms are given by (19.22.22). …
3: 13.3 Recurrence Relations and Derivatives
13.3.14 ( a + 1 ) z U ( a + 2 , b + 2 , z ) + ( z - b ) U ( a + 1 , b + 1 , z ) - U ( a , b , z ) = 0 .
4: 27.2 Functions
An equivalent form states that the n th prime p n (when the primes are listed in increasing order) is asymptotic to n ln n as n : …
5: Guide to Searching the DLMF
To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. …
6: 27.5 Inversion Formulas
The set of all number-theoretic functions f with f ( 1 ) 0 forms an abelian group under Dirichlet multiplication, with the function 1 / n in (27.2.5) as identity element; see Apostol (1976, p. 129). …For example, the equation ζ ( s ) ( 1 / ζ ( s ) ) = 1 is equivalent to the identity
27.5.2 d | n μ ( d ) = 1 n ,
7: 27.16 Cryptography
To code a piece x , raise x to the power r and reduce x r modulo n to obtain an integer y (the coded form of x ) between 1 and n . Thus, y x r ( mod n ) and 1 y < n . … By the Euler–Fermat theorem (27.2.8), x ϕ ( n ) 1 ( mod n ) ; hence x t ϕ ( n ) 1 ( mod n ) . But y s x r s x 1 + t ϕ ( n ) x ( mod n ) , so y s is the same as x modulo n . …
8: 30.3 Eigenvalues
With μ = m = 0 , 1 , 2 , , the spheroidal wave functions Ps n m ( x , γ 2 ) are solutions of Equation (30.2.1) which are bounded on ( - 1 , 1 ) , or equivalently, which are of the form ( 1 - x 2 ) 1 2 m g ( x ) where g ( z ) is an entire function of z . …
9: 28.2 Definitions and Basic Properties
The standard form of Mathieu’s equation with parameters ( a , q ) is …With ζ = cos z we obtain another algebraic form: … Equivalently, …If ν ^ = 0 or 1 , or equivalently, ν = n , then ν is a double root of the characteristic equation, otherwise it is a simple root. … An equivalent formulation is given by …
10: 23.20 Mathematical Applications
or equivalently, on replacing x by x / z and y by y / z (projective coordinates), into the form