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1: 14.30 Spherical and Spheroidal Harmonics
Distributional Completeness
2: 29.3 Definitions and Basic Properties
For each pair of values of ν and k there are four infinite unbounded sets of real eigenvalues h for which equation (29.2.1) has even or odd solutions with periods 2 K or 4 K . …
§29.3(ii) Distribution
In this table the nonnegative integer m corresponds to the number of zeros of each Lamé function in ( 0 , K ) , whereas the superscripts 2 m , 2 m + 1 , or 2 m + 2 correspond to the number of zeros in [ 0 , 2 K ) . …
0 K dn ( x , k ) ( Ec ν 2 m ( x , k 2 ) ) 2 d x = 1 4 π ,
To complete the definitions, Ec ν m ( K , k 2 ) is positive and d Es ν m ( z , k 2 ) / d z | z = K is negative. …
3: 27.2 Functions
27.2.1 n = r = 1 ν ( n ) p r a r ,
Tables of primes (§27.21) reveal great irregularity in their distribution. They tend to thin out among the large integers, but this thinning out is not completely regular. …
27.2.3 π ( x ) x ln x .
4: 22.4 Periods, Poles, and Zeros
§22.4(i) Distribution
Figure 22.4.1 illustrates the locations in the z -plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices 0 , 2 K , 2 K + 2 i K , 2 i K . … For the distribution of the k -zeros of the Jacobian elliptic functions see Walker (2009). … Figure 22.4.2 depicts the fundamental unit cell in the z -plane, with vertices s = 0 , c = K , d = K + i K , n = i K . The set of points z = m K + n i K , m , n , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by m K + n i K , where again m , n . …
5: Bibliography J
  • E. Jahnke, F. Emde, and F. Lösch (1966) Tafeln höherer Funktionen (Tables of Higher Functions). 7th edition, B. G. Teubner, Stuttgart (Bilingual).
  • A. T. James (1964) Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.
  • N. L. Johnson, S. Kotz, and N. Balakrishnan (1994) Continuous Univariate Distributions. 2nd edition, Vol. I, John Wiley & Sons Inc., New York.
  • N. L. Johnson, S. Kotz, and N. Balakrishnan (1995) Continuous Univariate Distributions. 2nd edition, Vol. II, John Wiley & Sons Inc., New York.
  • 6: 2.6 Distributional Methods
    §2.6 Distributional Methods
    Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … The Dirac delta distribution in (2.6.17) is given by … These equations again hold only in the sense of distributions. …
    §2.6(iv) Regularization
    7: Philip J. Davis
    He returned to Harvard after the war and completed a Ph. … He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. … Decades later, Olver became Editor-in-Chief and Mathematics Editor of the NIST Digital Library of Mathematical Functions (DLMF), a complete revision of A&S that was publicly released in 2010. …
    8: DLMF Project News
    error generating summary
    9: 28.12 Definitions and Basic Properties
    To complete the definition we require … To complete the definitions of the me ν functions we set …
    10: 1.16 Distributions
    1.16.30 D = ( 1 i x 1 , 1 i x 2 , , 1 i x n ) .