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1: 19.2 Definitions
Also, if k 2 and α 2 are real, then Π ( ϕ , α 2 , k ) is called a circular or hyperbolic case according as α 2 ( α 2 - k 2 ) ( α 2 - 1 ) is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points α 2 = 0 , k 2 , 1 . … Formulas involving Π ( ϕ , α 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ( x , y ) . …
2: 19.20 Special Cases
where x , y , z may be permuted. When the variables are real and distinct, the various cases of R J ( x , y , z , p ) are called circular (hyperbolic) cases if ( p - x ) ( p - y ) ( p - z ) is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. …
19.20.17 ( q + z ) R J ( 0 , y , z , - q ) = ( p - z ) R J ( 0 , y , z , p ) - 3 R F ( 0 , y , z ) , p = z ( y + q ) / ( z + q ) , w = z / ( z + q ) .
3: 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). …
4: 19.21 Connection Formulas
19.21.15 p R J ( 0 , y , z , p ) + q R J ( 0 , y , z , q ) = 3 R F ( 0 , y , z ) , p q = y z .
5: 19.6 Special Cases
Circular and hyperbolic cases, including Cauchy principal values, are unified by using R C ( x , y ) . …
6: 19.36 Methods of Computation
The step from n to n + 1 is an ascending Landen transformation if θ = 1 (leading ultimately to a hyperbolic case of R C ) or a descending Gauss transformation if θ = - 1 (leading to a circular case of R C ). …
7: 22.5 Special Values
In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
8: 13.20 Uniform Asymptotic Approximations for Large μ
(a) In the case - μ < κ < μ
13.20.9 ζ ζ 2 + α 2 + α 2 arcsinh ( ζ α ) = X μ - 2 κ μ ln ( X + x - 2 κ 2 μ 2 - κ 2 ) - 2 ln ( μ X + 2 μ 2 - κ x x μ 2 - κ 2 ) .
(b) In the case μ = κ (In both cases (a) and (b) the x -interval ( 0 , ) is mapped one-to-one onto the ζ -interval ( - , ) , with x = 0 and corresponding to ζ = - and , respectively.) …
13.20.13 ζ ζ 2 - α 2 - α 2 arccosh ( ζ α ) = X μ - 2 κ μ ln ( X + x - 2 κ 2 κ 2 - μ 2 ) - 2 ln ( κ x - μ X - 2 μ 2 x κ 2 - μ 2 ) , x 2 κ + 2 κ 2 - μ 2 ,
9: 4.43 Cubic Equations
4.43.2 z 3 + p z + q = 0
10: 28.23 Expansions in Series of Bessel Functions
28.23.3 me ν ( 0 , h 2 ) M ν ( j ) ( z , h ) = i tanh z n = - ( - 1 ) n ( ν + 2 n ) c 2 n ν ( h 2 ) 𝒞 ν + 2 n ( j ) ( 2 h cosh z ) ,
valid for all z when j = 1 , and for z > 0 and | cosh z | > 1 when j = 2 , 3 , 4 . …valid for all z when j = 1 , and for z > 0 and | sinh z | > 1 when j = 2 , 3 , 4 . In the case when ν is an integer …When j = 2 , 3 , 4 the series in the even-numbered equations converge for z > 0 and | cosh z | > 1 , and the series in the odd-numbered equations converge for z > 0 and | sinh z | > 1 . …