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11: 19.36 Methods of Computation
Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. …
12: 19.26 Addition Theorems
§19.26 Addition Theorems
§19.26(ii) Case x = 0
§19.26(iii) Duplication Formulas
19.26.20 R D ( x , y , z ) = 2 R D ( x + λ , y + λ , z + λ ) + 3 z ( z + λ ) .
19.26.21 2 R G ( x , y , z ) = 4 R G ( x + λ , y + λ , z + λ ) - λ R F ( x , y , z ) - x - y - z .
13: 3.2 Linear Algebra
Then | j k | 1 in all cases. … The Euclidean norm is the case p = 2 . … When A is a symmetric matrix, the left and right eigenvectors coincide, yielding κ ( λ ) = 1 , and the calculation of its eigenvalues is a well-conditioned problem.
§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix
Let A be an n × n symmetric matrix. …
14: 19.17 Graphics
§19.17 Graphics
See Figures 19.17.119.17.8 for symmetric elliptic integrals with real arguments. … For R F , R G , and R J , which are symmetric in x , y , z , we may further assume that z is the largest of x , y , z if the variables are real, then choose z = 1 , and consider only 0 x 1 and 0 y 1 . The cases x = 0 or y = 0 correspond to the complete integrals. The case y = 1 corresponds to elementary functions. …
15: 19.29 Reduction of General Elliptic Integrals
The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the 8 + 8 + 12 = 28 formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking x 2 as the variable of integration in 3. …
16: 19.22 Quadratic Transformations
Bartky’s Transformation
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. …
17: 19.7 Connection Formulas
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )
If k 2 and α 2 are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)). The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > csc 2 ϕ (see (19.6.5) for the complete case). …
18: 19.25 Relations to Other Functions
§19.25(i) Legendre’s Integrals as Symmetric Integrals
§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
§19.25(vii) Hypergeometric Function
19: 19.33 Triaxial Ellipsoids
19.33.1 S = 3 V R G ( a - 2 , b - 2 , c - 2 ) ,
§19.33(ii) Potential of a Charged Conducting Ellipsoid
19.33.5 V ( λ ) = Q R F ( a 2 + λ , b 2 + λ , c 2 + λ ) ,
19.33.6 1 / C = R F ( a 2 , b 2 , c 2 ) .
A conducting elliptic disk is included as the case c = 0 . …
20: 19.18 Derivatives and Differential Equations
The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). …