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11: 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
12: 3.5 Quadrature
Similar results hold for the trapezoidal rule in the formIf f C 2 m + 2 [ a , b ] , then the remainder E n ( f ) in (3.5.2) can be expanded in the formThe p n ( x ) are the monic Legendre polynomials, that is, the polynomials P n ( x ) 18.3) scaled so that the coefficient of the highest power of x in their explicit forms is unity. … Integrals of the formThe integral (3.5.39) has the form (3.5.35) if we set ζ = t p , c = t σ , and f ( ζ ) = t - 1 ζ s G ( ζ / t ) . …
13: 28.8 Asymptotic Expansions for Large q
28.8.9 W m ± ( x ) = e ± 2 h sin x ( cos x ) m + 1 { ( cos ( 1 2 x + 1 4 π ) ) 2 m + 1 , ( sin ( 1 2 x + 1 4 π ) ) 2 m + 1 ,
28.8.11 P m ( x ) 1 + s 2 3 h cos 2 x + 1 h 2 ( s 4 + 86 s 2 + 105 2 11 cos 4 x - s 4 + 22 s 2 + 57 2 11 cos 2 x ) + ,
28.8.12 Q m ( x ) sin x cos 2 x ( 1 2 5 h ( s 2 + 3 ) + 1 2 9 h 2 ( s 3 + 3 s + 4 s 3 + 44 s cos 2 x ) ) + .
14: 28.20 Definitions and Basic Properties
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 - 1 ) w ′′ + ζ w + ( 4 q ζ 2 - 2 q - a ) w = 0 , ζ = cosh z .
15: 36.7 Zeros
There are also three sets of zero lines in the plane z = 0 related by 2 π / 3 rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates ( x = r cos θ , y = r sin θ ) is given by …
16: 8.21 Generalized Sine and Cosine Integrals
8.21.18 f ( a , z ) = si ( a , z ) cos z - ci ( a , z ) sin z ,
8.21.19 g ( a , z ) = si ( a , z ) sin z + ci ( a , z ) cos z .
8.21.22 f ( a , z ) = 0 sin t ( t + z ) 1 - a d t ,
8.21.23 g ( a , z ) = 0 cos t ( t + z ) 1 - a d t .
17: 4.37 Inverse Hyperbolic Functions
For the corresponding results for arccsch z , arcsech z , and arccoth z , use (4.37.7)–(4.37.9); compare §4.23(iv). …
18: 14.5 Special Values
14.5.22 Q 1 2 ( cos θ ) = K ( cos ( 1 2 θ ) ) - 2 E ( cos ( 1 2 θ ) ) ,
14.5.23 Q - 1 2 ( cos θ ) = K ( cos ( 1 2 θ ) ) .
14.5.26 Q 1 2 ( cosh ξ ) = 2 π - 1 / 2 cosh ξ sech ( 1 2 ξ ) K ( sech ( 1 2 ξ ) ) - 4 π - 1 / 2 cosh ( 1 2 ξ ) E ( sech ( 1 2 ξ ) ) ,
19: 28.4 Fourier Series
28.4.1 ce 2 n ( z , q ) = m = 0 A 2 m 2 n ( q ) cos 2 m z ,
20: 9.5 Integral Representations
9.5.1 Ai ( x ) = 1 π 0 cos ( 1 3 t 3 + x t ) d t .
9.5.2 Ai ( - x ) = x 1 / 2 π - 1 cos ( x 3 / 2 ( 1 3 t 3 + t 2 - 2 3 ) ) d t , x > 0 .