# locally integrable

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##### 1: 2.6 Distributional Methods
Let $f(t)$ be locally integrable on $[0,\infty)$. The Stieltjes transform of $f(t)$ is defined by …Since $f(t)$ is locally integrable on $[0,\infty)$, it defines a distribution by … In terms of the convolution product …of two locally integrable functions on $[0,\infty)$, (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 $(0,\infty)$, that is, $\int_{\rho}^{T}f(t)\,\mathrm{d}t$ exists for all $\rho$ and $T$ satisfying $0<\rho. … Let $f(t)$ and $h(t)$ be locally integrable on $(0,\infty)$ and …Also, let …
2.5.24 $h_{2}(t)=h(t)-h_{1}(t).$
##### 3: 1.16 Distributions
A distribution $\Lambda$ is called regular if there is a locally integrable function $f$ on $I$ (i. … If $f$ is a locally integrable function then its distributional derivative is $Df=\Lambda^{\prime}_{f}$. … A locally integrable function $f(x)=f(x_{1},x_{2},\dots,x_{n})$ gives rise to a distribution $\Lambda_{f}$ defined by …
##### 4: 9.13 Generalized Airy Functions
9.13.25 $A_{k}\left(z,p\right)=\frac{1}{2\pi i}\int_{\mathscr{L}_{k}}t^{-p}\exp\left(zt% -\tfrac{1}{3}t^{3}\right)\,\mathrm{d}t,$ $k=1,2,3$, $p\in\mathbb{C}$,
##### 5: 28.28 Integrals, Integral Representations, and Integral Equations
In (28.28.7)–(28.28.9) the paths of integration $\mathcal{L}_{j}$ are given by …
28.28.7 $\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}e^{2\mathrm{i}hw}\operatorname{me}_{\nu}% \left(t,h^{2}\right)\,\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\operatorname{me}_{% \nu}\left(\alpha,h^{2}\right){\operatorname{M}^{(j)}_{\nu}}\left(z,h\right),$ $j=3,4$,
28.28.9 $\dfrac{1}{2\pi}\int_{\mathcal{L}_{1}}e^{2\mathrm{i}hw}\operatorname{me}_{\nu}% \left(t,h^{2}\right)\,\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\operatorname{me}_{% \nu}\left(\alpha,h^{2}\right){\operatorname{M}^{(1)}_{\nu}}\left(z,h\right).$
28.28.33 $\gamma_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\operatorname{me}_{\nu}'\left(t% \right)\operatorname{me}_{-\nu-2m}\left(t\right)\,\mathrm{d}t=(-1)^{m}\dfrac{4% \mathrm{i}}{\pi}\frac{\operatorname{me}_{\nu}'\left(0\right)\operatorname{me}_% {-\nu-2m}\left(0\right)}{\operatorname{D}_{1}\left(\nu,\nu+2m,0\right)}.$
28.28.43 $\widehat{\beta}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\sin t\operatorname{se}_{n% }\left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)\,\mathrm{d}t=(-% 1)^{p}\dfrac{2}{\mathrm{i}\pi}\dfrac{\operatorname{se}_{n}'\left(0,h^{2}\right% )\operatorname{ce}_{m}\left(0,h^{2}\right)}{h\operatorname{Dsc}_{1}\left(n,m,0% \right)}.$
##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Thus, and this is a case where $q(x)$ is not continuous, if $q(x)=-\alpha\delta\left(x-a\right)$, $\alpha>0$, there will be an $L^{2}$ eigenfunction localized in the vicinity of $x=a$, with a negative eigenvalue, thus disjoint from the continuous spectrum on $[0,\infty)$. …
##### 7: 36.12 Uniform Approximation of Integrals
In the cuspoid case (one integration variable)
36.12.6 $A(\mathbf{y})=f(u(0,\mathbf{y});\mathbf{y}),$
36.12.8 $a_{m}(\mathbf{y})=\sum_{n=1}^{K+1}\frac{P_{mn}(\mathbf{y})G_{n}(\mathbf{y})}{(% t_{n}(\mathbf{x}(\mathbf{y})))^{m+1}\prod\limits_{\begin{subarray}{c}l=1\\ l\neq n\end{subarray}}^{K+1}(t_{n}(\mathbf{x}(\mathbf{y}))-t_{l}(\mathbf{x}(% \mathbf{y})))},$
##### 8: 32.2 Differential Equations
be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $\mathbb{C}$. 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 $\phi(z)$ are locally analytic functions. …
##### 9: 31.9 Orthogonality
The integration path begins at $z=\zeta$, encircles $z=1$ once in the positive sense, followed by $z=0$ once in the positive sense, and so on, returning finally to $z=\zeta$. The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). …
$f_{0}(q_{m},z)=\mathit{H\!\ell}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$
$f_{1}(q_{m},z)=\mathit{H\!\ell}\left(1-a,\alpha\beta-q_{m};\alpha,\beta,\delta% ,\gamma;1-z\right),$
and the integration paths $\mathcal{L}_{1}$, $\mathcal{L}_{2}$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …
##### 10: 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 $\Phi^{(\alpha,\beta)}_{\lambda}(t)=(2\cosh t)^{\mathrm{i}\lambda-\alpha-\beta-% 1}F\left({\tfrac{1}{2}(\alpha+\beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha-% \beta+1-\mathrm{i}\lambda)\atop 1-\mathrm{i}\lambda};{\operatorname{sech}}^{2}% t\right).$