About the Project

with respect to integration

AdvancedHelp

(0.005 seconds)

1—10 of 63 matching pages

1: 19.13 Integrals of Elliptic Integrals
§19.13(i) Integration with Respect to the Modulus
§19.13(ii) Integration with Respect to the Amplitude
2: 1.1 Special Notation
x , y real variables.
L 2 ( X , d α ) the space of all Lebesgue–Stieltjes measurable functions on X which are square integrable with respect to d α .
3: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
4: 3.11 Approximation Techniques
They enjoy an orthogonal property with respect to integrals: …
5: 1.4 Calculus of One Variable
Stieltjes integrability for f with respect to α can be defined similarly as Riemann integrability in the case that α ( x ) is differentiable with respect to x ; a generalization follows below. … Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure d μ ( x ) and it is well defined for functions f which are integrable with respect to that more general measure. …
1.4.23_2 a b f ( x ) d α ( x ) = a b f ( x ) w ( x ) d x , f integrable with respectto d α .
6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
For a Lebesgue–Stieltjes measure d α on X let L 2 ( X , d α ) be the space of all Lebesgue–Stieltjes measurable complex-valued functions on X which are square integrable with respect to d α , …
7: 36.2 Catastrophes and Canonical Integrals
with the contour passing to the lower right of u = 0 . …with the contour passing to the upper right of u = 0 . … Ψ 1 is related to the Airy function (§9.2): … … Addendum: For further special cases see §36.2(iv)
8: 20.10 Integrals
§20.10(i) Mellin Transforms with respect to the Lattice Parameter
20.10.1 0 x s 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 2 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.2 0 x s 1 ( θ 3 ( 0 | i x 2 ) 1 ) d x = π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.3 0 x s 1 ( 1 θ 4 ( 0 | i x 2 ) ) d x = ( 1 2 1 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
9: 18.18 Sums
In all three cases of Jacobi, Laguerre and Hermite, if f ( x ) is L 2 on the corresponding interval with respect to the corresponding weight function and if a n , b n , d n are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in L 2 sense. …
10: 1.17 Integral and Series Representations of the Dirac Delta
In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non- L 2 improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. …