About the Project
NIST

closed

AdvancedHelp

(0.001 seconds)

1—10 of 139 matching pages

1: 7.23 Tables
  • Abramowitz and Stegun (1964, Chapter 7) includes erf x , ( 2 / π ) e - x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) e - x 2 , x [ 2 , 10 ] , 8S; x e x 2 erfc x , x - 2 [ 0 , 0.25 ] , 7D; 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 1 ( 1 ) 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ( x ) , x [ 0 , 2 ] , 10D; x F ( x ) , x - 2 [ 0 , 0.25 ] , 9D; C ( x ) , S ( x ) , x [ 0 , 5 ] , 7D; f ( x ) , g ( x ) , x [ 0 , 1 ] , x - 1 [ 0 , 1 ] , 15D.

  • Finn and Mugglestone (1965) includes the Voigt function H ( a , u ) , u [ 0 , 22 ] , a [ 0 , 1 ] , 6S.

  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • 2: 29.9 Stability
    If ν is not an integer, then (29.2.1) is unstable iff h a ν 0 ( k 2 ) or h lies in one of the closed intervals with endpoints a ν m ( k 2 ) and b ν m ( k 2 ) , m = 1 , 2 , . If ν is a nonnegative integer, then (29.2.1) is unstable iff h a ν 0 ( k 2 ) or h [ b ν m ( k 2 ) , a ν m ( k 2 ) ] for some m = 1 , 2 , , ν .
    3: 22.17 Moduli Outside the Interval [0,1]
    §22.17 Moduli Outside the Interval [0,1]
    Jacobian elliptic functions with real moduli in the intervals ( - , 0 ) and ( 1 , ) , or with purely imaginary moduli are related to functions with moduli in the interval [ 0 , 1 ] by the following formulas. … For proofs of these results and further information see Walker (2003).
    4: Mathematical Introduction
    complex plane (excluding infinity).
    f ( z ) | C = 0 f ( z ) is continuous at all points of a simple closed contour C in .
    [ a , b ] closed interval in , or closed straight-line segment joining a and b in .
    ( a , b ] or [ a , b ) half-closed intervals.
    5: About Color Map
    Mathematically, we scale the height to h lying in the interval [ 0 , 4 ] and the components are computed as follows … Specifically, by scaling the phase angle in [ 0 , 2 π ) to q in the interval [ 0 , 4 ) , the hue (in degrees) is computed as …
    6: 4.37 Inverse Hyperbolic Functions
    It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on ( - , 1 ] . …
    4.37.22 arccosh x = ± ln ( i ( 1 - x 2 ) 1 / 2 + x ) , x ( - 1 , 1 ] ,
    7: 5.14 Multidimensional Integrals
    5.14.4 [ 0 , 1 ] n t 1 t 2 t m | Δ ( t 1 , , t n ) | 2 c k = 1 n t k a - 1 ( 1 - t k ) b - 1 d t k = 1 ( Γ ( 1 + c ) ) n k = 1 m a + ( n - k ) c a + b + ( 2 n - k - 1 ) c k = 1 n Γ ( a + ( n - k ) c ) Γ ( b + ( n - k ) c ) Γ ( 1 + k c ) Γ ( a + b + ( 2 n - k - 1 ) c ) ,
    5.14.5 [ 0 , ) n t 1 t 2 t m | Δ ( t 1 , , t n ) | 2 c k = 1 n t k a - 1 e - t k d t k = k = 1 m ( a + ( n - k ) c ) k = 1 n Γ ( a + ( n - k ) c ) Γ ( 1 + k c ) ( Γ ( 1 + c ) ) n ,
    5.14.7 1 ( 2 π ) n [ - π , π ] n 1 j < k n | e i θ j - e i θ k | 2 b d θ 1 d θ n = Γ ( 1 + b n ) ( Γ ( 1 + b ) ) n , b > - 1 / n .
    8: 22.18 Mathematical Applications
    With k [ 0 , 1 ] the mapping z w = sn ( z , k ) gives a conformal map of the closed rectangle [ - K , K ] × [ 0 , K ] onto the half-plane w 0 , with 0 , ± K , ± K + i K , i K mapping to 0 , ± 1 , ± k - 2 , respectively. The half-open rectangle ( - K , K ) × [ - K , K ] maps onto cut along the intervals ( - , - 1 ] and [ 1 , ) . …
    9: 7.20 Mathematical Applications
    See accompanying text
    Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by x = C ( t ) , y = S ( t ) , t [ 0 , ) . Magnify
    10: 18.24 Hahn Class: Asymptotic Approximations
    With μ = N / n and x fixed, Qiu and Wong (2004) gives an asymptotic expansion for K n ( x ; p , N ) as n , that holds uniformly for μ [ 1 , ) . … Taken together, these expansions are uniformly valid for - < x < and for a in unbounded intervals—each of which contains [ 0 , ( 1 - δ ) n ] , where δ again denotes an arbitrary small positive constant. … These approximations are in terms of Laguerre polynomials and hold uniformly for ph ( x + i λ ) [ 0 , π ] . …