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11: 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
12: 35.2 Laplace Transform
35.2.1 g ( Z ) = Ω etr ( - Z X ) f ( X ) d X ,
where the integration variable X ranges over the space Ω . …
35.2.3 f 1 * f 2 ( T ) = 0 < X < T f 1 ( T - X ) f 2 ( X ) d X .
13: 6.16 Mathematical Applications
6.16.2 S n ( x ) = k = 0 n - 1 sin ( ( 2 k + 1 ) x ) 2 k + 1 = 1 2 0 x sin ( 2 n t ) sin t d t = 1 2 Si ( 2 n x ) + R n ( x ) ,
6.16.3 R n ( x ) = 1 2 0 x ( 1 sin t - 1 t ) sin ( 2 n t ) d t .
By integration by parts …
14: 2.3 Integrals of a Real Variable
§2.3(i) Integration by Parts
Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … derives from the neighborhood of the minimum of p ( t ) in the integration range. … A uniform approximation can be constructed by quadratic change of integration variable: … We replace the limit κ by and integrate term-by-term: …
15: 1.14 Integral Transforms
In many applications f ( t ) is absolutely integrable and f ( t ) is continuous on ( - , ) . … Suppose f ( t ) and g ( t ) are absolutely integrable on ( - , ) , then …
Differentiation and Integration
Suppose x - σ f ( x ) and x σ - 1 g ( x ) are absolutely integrable on ( 0 , ) and either g ( σ + i t ) or f ( 1 - σ - i t ) is absolutely integrable on ( - , ) . … If f ( t ) is absolutely integrable on [ 0 , R ] for every finite R , and the integral (1.14.47) converges, then …
16: 8.21 Generalized Sine and Cosine Integrals
(obtained from (5.2.1) by rotation of the integration path) is also needed. … In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. …
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 .
17: 21.7 Riemann Surfaces
21.7.2 P ~ ( λ ~ , μ ~ , η ~ ) = 0 ,
If a local coordinate z is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by …Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles a j , b j , j = 1 , 2 , , g . … where P 1 and P 2 are points on Γ , ω = ( ω 1 , ω 2 , , ω g ) , and the path of integration on Γ from P 1 to P 2 is identical for all components. … where again all integration paths are identical for all components. …
18: 3.11 Approximation Techniques
3.11.3 m j = ( - 1 ) j ϵ n ( x j ) , j = 0 , 1 , , n + 1 .
The iterative process converges locally and quadratically (§3.8(i)). … They enjoy an orthogonal property with respect to integrals: …
3.11.16 R k , ( x ) = p 0 + p 1 x + + p k x k 1 + q 1 x + + q x
3.11.20 f ( z ) = c 0 + c 1 z + c 2 z 2 +
19: 30.4 Functions of the First Kind
If f ( x ) is mean-square integrable on [ - 1 , 1 ] , then formally …
30.4.8 c n = ( n + 1 2 ) ( n - m ) ! ( n + m ) ! - 1 1 f ( t ) Ps n m ( t , γ 2 ) d t .
20: 2.4 Contour Integrals
If q ( t ) is analytic in a sector α 1 < ph t < α 2 containing ph t = 0 , then the region of validity may be increased by rotation of the integration paths. … On the interval 0 < t < let q ( t ) be differentiable and e - c t q ( t ) be absolutely integrable, where c is a real constant. … Then by integration by parts the integral … The most successful results are obtained on moving the integration contour as far to the left as possible. … The change of integration variable is given by …