# with respect to integration

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##### 2: 1.1 Special Notation
 $x,y$ real variables. … the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$. …
##### 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 $\alpha$ can be defined similarly as Riemann integrability in the case that $\alpha(x)$ 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 $\,\mathrm{d}\mu(x)$ and it is well defined for functions $f$ which are integrable with respect to that more general measure. …
1.4.23_2 $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}f(x)w(x)\,\mathrm{d}x,$ $f$ integrable with respectto $\,\mathrm{d}\alpha$.
##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
For a Lebesgue–Stieltjes measure $\,\mathrm{d}\alpha$ on $X$ let $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$, …
##### 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 $u=0$. … $\Psi_{1}$ 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 respectto the Lattice Parameter
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$,
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$,
20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>0$.
##### 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. …