# over infinite intervals

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## 1—10 of 18 matching pages

##### 1: 13.4 Integral Representations
13.4.2 ${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(b-c\right)}\int_{0}^{1}{% \mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-1}\,\mathrm{d}t,$ $\Re b>\Re c>0$,
13.4.6 $U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{\Gamma\left(1+a-b\right)}\int_{0}% ^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t\right)e^{-t}t^{b+n-1}}{t+z}\,\mathrm{% d}t,$ $\left|\operatorname{ph}z\right|<\pi$, $n=0,1,2,\dots$, $-\Re b,
##### 3: 13.16 Integral Representations
13.16.2 $M_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(1+2\mu\right)z^{\lambda}}{% \Gamma\left(1+2\mu-2\lambda\right)\Gamma\left(2\lambda\right)}\*\int_{0}^{1}M_% {\kappa-\lambda,\mu-\lambda}\left(zt\right)e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda% -\frac{1}{2}}{(1-t)^{2\lambda-1}}\,\mathrm{d}t,$ $\Re\mu+\tfrac{1}{2}>\Re\lambda>0$,
##### 4: 9.12 Scorer Functions
If $\zeta=\tfrac{2}{3}z^{3/2}$ or $\tfrac{2}{3}x^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then …
##### 5: 36.9 Integral Identities
For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). …
##### 6: 10.22 Integrals
###### §10.22(iii) Integrals over the Interval$(x,\infty)$
Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …
##### 7: 2.8 Differential Equations with a Parameter
and for simplicity $\xi$ is assumed to range over a finite or infinite interval $(\alpha_{1},\alpha_{2})$ with $\alpha_{1}<0$, $\alpha_{2}>0$. …
##### 8: 18.39 Applications in the Physical Sciences
where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\hbar=h/(2\pi)$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval. …
##### 9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Two elements $u$ and $v$ in $V$ are orthogonal if $\left\langle u,v\right\rangle=0$. … where the infinite sum means convergence in norm, … Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $\mathbb{R}$. … Let $X=(a,b)$ be a finite or infinite open interval in $\mathbb{R}$. … We assume a continuous spectrum $\lambda\in\boldsymbol{\sigma}_{c}=[0,\infty)$, and a finite or countably infinite point spectrum $\boldsymbol{\sigma}_{p}$ with elements $\lambda_{n}$. …
##### 10: 2.3 Integrals of a Real Variable
If, in addition, $q(t)$ is infinitely differentiable on $[0,\infty)$ and … assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. …Alternatively, assume $b=\infty$, $q(t)$ is infinitely differentiable on $[a,\infty)$, and each of the integrals $\int e^{ixt}q^{(s)}(t)\,\mathrm{d}t$, $s=0,1,2,\dots$, converges as $t\to\infty$ uniformly for all sufficiently large $x$. … Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\infty)$, and each of the integrals $\int e^{ixt}q^{(s)}(t)\,\mathrm{d}t$, $s=0,1,2,\dots$, converges at $t=\infty$, uniformly for all sufficiently large $x$. …
• (a)

On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and $p^{\prime}(t)>0$.