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1: 2.6 Distributional Methods
Let f ( t ) be locally integrable on [ 0 , ) . The Stieltjes transform of f ( t ) is defined by …Since f ( t ) is locally integrable on [ 0 , ) , it defines a distribution by … In terms of the convolution product …of two locally integrable functions on [ 0 , ) , (2.6.33) can be written …
2: 2.5 Mellin Transform Methods
§2.5(i) Introduction
Let f ( t ) be a locally integrable function on ( 0 , ) , that is, ρ T f ( t ) d t exists for all ρ and T satisfying 0 < ρ < T < . … Let f ( t ) and h ( t ) be locally integrable on ( 0 , ) and …Also, let …
2.5.24 h 2 ( t ) = h ( t ) - h 1 ( t ) .
3: 1.16 Distributions
For any locally integrable function f , its distributional derivative is D f = Λ f . … A locally integrable function f ( x ) = f ( x 1 , x 2 , , x n ) gives rise to a distribution Λ f defined by …
4: 9.13 Generalized Airy Functions
9.13.25 A k ( z , p ) = 1 2 π i k t - p exp ( z t - 1 3 t 3 ) d t , k = 1 , 2 , 3 , p ,
5: 28.28 Integrals, Integral Representations, and Integral Equations
In (28.28.7)–(28.28.9) the paths of integration j are given by …
28.28.7 1 π j e 2 i h w me ν ( t , h 2 ) d t = e i ν π / 2 me ν ( α , h 2 ) M ν ( j ) ( z , h ) , j = 3 , 4 ,
28.28.33 γ ν , m = 1 2 π 0 2 π me ν ( t ) me - ν - 2 m ( t ) d t = ( - 1 ) m 4 i π me ν ( 0 ) me - ν - 2 m ( 0 ) D 1 ( ν , ν + 2 m , 0 ) .
28.28.43 β ^ n , m = 1 2 π 0 2 π sin t se n ( t , h 2 ) ce m ( t , h 2 ) d t = ( - 1 ) p 2 i π se n ( 0 , h 2 ) ce m ( 0 , h 2 ) h Dsc 1 ( n , m , 0 ) .
6: 36.12 Uniform Approximation of Integrals
In the cuspoid case (one integration variable)
36.12.1 I ( y , k ) = - exp ( i k f ( u ; y ) ) g ( u , y ) d u ,
36.12.4 f ( u ( t , y ) ; y ) = A ( y ) + Φ K ( t ; x ( y ) ) ,
36.12.6 A ( y ) = f ( u ( 0 , y ) ; y ) ,
36.12.8 a m ( y ) = n = 1 K + 1 P m n ( y ) G n ( y ) ( t n ( x ( y ) ) ) m + 1 l = 1 l n K + 1 ( t n ( x ( y ) ) - t l ( x ( y ) ) ) ,
7: 32.2 Differential Equations
be a nonlinear second-order differential equation in which F is a rational function of w and d w / d z , and is locally analytic in z , that is, analytic except for isolated singularities in . In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. … in which a ( z ) , b ( z ) , c ( z ) , d ( z ) , and ϕ ( z ) are locally analytic functions. …
8: 31.9 Orthogonality
The integration path begins at z = ζ , encircles z = 1 once in the positive sense, followed by z = 0 once in the positive sense, and so on, returning finally to z = ζ . The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). …
f 0 ( q m , z ) = H ( a , q m ; α , β , γ , δ ; z ) ,
f 1 ( q m , z ) = H ( 1 - a , α β - q m ; α , β , δ , γ ; 1 - z ) ,
and the integration paths 1 , 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . …
9: 15.9 Relations to Other Functions
where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and
15.9.14 Φ λ ( α , β ) ( t ) = ( 2 cosh t ) i λ - α - β - 1 F ( 1 2 ( α + β + 1 - i λ ) , 1 2 ( α - β + 1 - i λ ) 1 - i λ ; sech 2 t ) .
10: 3.5 Quadrature
§3.5 Quadrature
§3.5(iii) Romberg Integration
Further refinements are achieved by Romberg integration. … For these cases the integration path may need to be deformed; see §3.5(ix). …
§3.5(ix) Other Contour Integrals