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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: 14.27 Zeros
    P ν μ ( x ± i 0 ) (either side of the cut) has exactly one zero in the interval ( , 1 ) if either of the following sets of conditions holds: …For all other values of the parameters P ν μ ( x ± i 0 ) has no zeros in the interval ( , 1 ) . …
    3: 26.15 Permutations: Matrix Notation
    If ( j , k ) B , then σ ( j ) k . The number of derangements of n is the number of permutations with forbidden positions B = { ( 1 , 1 ) , ( 2 , 2 ) , , ( n , n ) } . … For ( j , k ) B , B [ j , k ] denotes B after removal of all elements of the form ( j , t ) or ( t , k ) , t = 1 , 2 , , n . B ( j , k ) denotes B with the element ( j , k ) removed. … Let B = { ( j , j ) , ( j , j + 1 ) |  1 j < n } { ( n , n ) , ( n , 1 ) } . …
    4: 14.16 Zeros
    §14.16(ii) Interval 1 < x < 1
    The zeros of 𝖰 ν μ ( x ) in the interval ( 1 , 1 ) interlace those of 𝖯 ν μ ( x ) . …
    §14.16(iii) Interval 1 < x <
    P ν μ ( x ) has exactly one zero in the interval ( 1 , ) if either of the following sets of conditions holds: … 𝑸 ν μ ( x ) has no zeros in the interval ( 1 , ) when ν > 1 , and at most one zero in the interval ( 1 , ) when ν < 1 .
    5: 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).
    6: 1.4 Calculus of One Variable
    Suppose f ( x ) is defined on [ a , b ] . … A function f ( x ) is convex on ( a , b ) if …for any c , d ( a , b ) , and t [ 0 , 1 ] . …A similar definition applies to closed intervals [ a , b ] . …
    7: 26.6 Other Lattice Path Numbers
    D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . … r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
    8: 4.37 Inverse Hyperbolic Functions
    In (4.37.2) the integration path may not pass through either of the points ± 1 , and the function ( t 2 1 ) 1 / 2 assumes its principal value when t ( 1 , ) . … 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 ] ,
    9: 1.5 Calculus of Two or More Variables
    A function f ( x , y ) is continuous at a point ( a , b ) if … f ( x , y ) has a local minimum (maximum) at ( a , b ) if … for all c 1 [ c 0 , d ) and all x [ a , b ] . … Let f ( x , y ) be defined on a closed rectangle R = [ a , b ] × [ c , d ] . … Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , then …
    10: 4.3 Graphics
    See accompanying text
    Figure 4.3.1: ln x and e x . Parallel tangent lines at ( 1 , 0 ) and ( 0 , 1 ) make evident the mirror symmetry across the line y = x , demonstrating the inverse relationship between the two functions. Magnify
    In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . …