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11: 21.1 Special Notation
g , h positive integers.
𝛀 g × g complex, symmetric matrix with 𝛀 strictly positive definite, i.e., a Riemann matrix.
12: 18.16 Zeros
Then θ n , m is strictly increasing in α and strictly decreasing in β ; furthermore, if α = β , then θ n , m is strictly increasing in α . …
13: 8.13 Zeros
The negative zero x ( a ) decreases monotonically in the interval 1 < a < 0 , and satisfies …
14: 9.8 Modulus and Phase
§9.8(iii) Monotonicity
15: 26.9 Integer Partitions: Restricted Number and Part Size
equivalently, partitions into at most k parts either have exactly k parts, in which case we can subtract one from each part, or they have strictly fewer than k parts. …
16: 8.3 Graphics
Some monotonicity properties of γ ( a , x ) and Γ ( a , x ) in the four quadrants of the ( a , x )-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). …
17: 18.36 Miscellaneous Polynomials
For the Laguerre polynomials L n ( α ) ( x ) this requires, omitting all strictly positive factors, …
18: 22.20 Methods of Computation
If both k and x are real then dn is strictly positive and dn ( x , k ) = 1 k 2 sn 2 ( x , k ) which follows from (22.6.1). …
19: 3.8 Nonlinear Equations
  • (a)

    f ( x 0 ) f ′′ ( x 0 ) > 0 and f ( x ) , f ′′ ( x ) do not change sign between x 0 and ξ (monotonic convergence).

  • (b)

    f ( x 0 ) f ′′ ( x 0 ) < 0 , f ( x ) , f ′′ ( x ) do not change sign in the interval ( x 0 , x 1 ) , and ξ [ x 0 , x 1 ] (monotonic convergence after the first iteration).

  • 20: 1.11 Zeros of Polynomials
    with real coefficients, is called stable if the real parts of all the zeros are strictly negative. …