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1: 18.8 Differential Equations
Table 18.8.1: Classical OP’s: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
f ( x ) A ( x ) B ( x ) C ( x ) λ n
C n ( λ ) ( x ) 1 - x 2 - ( 2 λ + 1 ) x 0 n ( n + 2 λ )
T n ( x ) 1 - x 2 - x 0 n 2
P n ( x ) 1 - x 2 - 2 x 0 n ( n + 1 )
H n ( x ) 1 - 2 x 0 2 n
He n ( x ) 1 - x 0 n
2: 18.31 Bernstein–Szegő Polynomials
Let ρ ( x ) be a polynomial of degree and positive when - 1 x 1 . The Bernstein–Szegő polynomials { p n ( x ) } , n = 0 , 1 , , are orthogonal on ( - 1 , 1 ) with respect to three types of weight function: ( 1 - x 2 ) - 1 2 ( ρ ( x ) ) - 1 , ( 1 - x 2 ) 1 2 ( ρ ( x ) ) - 1 , ( 1 - x ) 1 2 ( 1 + x ) - 1 2 ( ρ ( x ) ) - 1 . …
3: 4.5 Inequalities
4.5.1 x 1 + x < ln ( 1 + x ) < x , x > - 1 , x 0 ,
4.5.2 x < - ln ( 1 - x ) < x 1 - x , x < 1 , x 0 ,
4.5.7 e - x / ( 1 - x ) < 1 - x < e - x , x < 1 ,
4.5.11 x < e x - 1 < x 1 - x , x < 1 ,
4.5.12 e x / ( 1 + x ) < 1 + x , x > - 1 ,
4: 6.8 Inequalities
6.8.1 1 2 ln ( 1 + 2 x ) < e x E 1 ( x ) < ln ( 1 + 1 x ) ,
6.8.2 x x + 1 < x e x E 1 ( x ) < x + 1 x + 2 ,
6.8.3 x ( x + 3 ) x 2 + 4 x + 2 < x e x E 1 ( x ) < x 2 + 5 x + 2 x 2 + 6 x + 6 .
5: 14.10 Recurrence Relations and Derivatives
14.10.1 P ν μ + 2 ( x ) + 2 ( μ + 1 ) x ( 1 - x 2 ) - 1 / 2 P ν μ + 1 ( x ) + ( ν - μ ) ( ν + μ + 1 ) P ν μ ( x ) = 0 ,
14.10.2 ( 1 - x 2 ) 1 / 2 P ν μ + 1 ( x ) - ( ν - μ + 1 ) P ν + 1 μ ( x ) + ( ν + μ + 1 ) x P ν μ ( x ) = 0 ,
14.10.4 ( 1 - x 2 ) d P ν μ ( x ) d x = ( μ - ν - 1 ) P ν + 1 μ ( x ) + ( ν + 1 ) x P ν μ ( x ) ,
14.10.6 P ν μ + 2 ( x ) + 2 ( μ + 1 ) x ( x 2 - 1 ) - 1 / 2 P ν μ + 1 ( x ) - ( ν - μ ) ( ν + μ + 1 ) P ν μ ( x ) = 0 ,
14.10.7 ( x 2 - 1 ) 1 / 2 P ν μ + 1 ( x ) - ( ν - μ + 1 ) P ν + 1 μ ( x ) + ( ν + μ + 1 ) x P ν μ ( x ) = 0 .
6: 12.6 Continued Fraction
For a continued-fraction expansion of the ratio U ( a , x ) / U ( a - 1 , x ) see Cuyt et al. (2008, pp. 340–341).
7: 4.12 Generalized Logarithms and Exponentials
4.12.1 ϕ ( x + 1 ) = e ϕ ( x ) , - 1 < x < ,
and is strictly increasing when 0 x 1 . …It, too, is strictly increasing when 0 x 1 , and …
4.12.5 ϕ ( x ) = ψ ( x ) = x , 0 x 1 .
4.12.6 ϕ ( x ) = ln ( x + 1 ) , - 1 < x < 0 ,
8: 27.7 Lambert Series as Generating Functions
27.7.1 n = 1 f ( n ) x n 1 - x n .
If | x | < 1 , then the quotient x n / ( 1 - x n ) is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: … Again with | x | < 1 , special cases of (27.7.2) include:
27.7.3 n = 1 μ ( n ) x n 1 - x n = x ,
27.7.4 n = 1 ϕ ( n ) x n 1 - x n = x ( 1 - x ) 2 ,
9: 19.32 Conformal Map onto a Rectangle
with x 1 , x 2 , x 3 real constants, has differential …
19.32.3 x 1 > x 2 > x 3 ,
z ( x 1 ) = R F ( 0 , x 1 - x 2 , x 1 - x 3 ) ( > 0 ) ,
z ( x 2 ) = z ( x 1 ) + z ( x 3 ) ,
z ( x 3 ) = R F ( x 3 - x 1 , x 3 - x 2 , 0 ) = - i R F ( 0 , x 1 - x 3 , x 2 - x 3 ) .
10: 20.3 Graphics
See accompanying text
Figure 20.3.6: θ 1 ( x , q ) , 0 q 1 , x = 0, 0. … Magnify
See accompanying text
Figure 20.3.7: θ 2 ( x , q ) , 0 q 1 , x = 0, 0. … Magnify
See accompanying text
Figure 20.3.8: θ 3 ( x , q ) , 0 q 1 , x = 0, 0. … Magnify
See accompanying text
Figure 20.3.9: θ 4 ( x , q ) , 0 q 1 , x = 0, 0. … Magnify
See accompanying text
Figure 20.3.14: θ 1 ( π x + i y , 0.12 ) , - 1 x 1 , - 1 y 2.3 . Magnify 3D Help