# over infinite intervals

(0.003 seconds)

## 1—10 of 16 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: 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 $p^{\prime}(t)>0$.

• ##### 9: 1.4 Calculus of One Variable
If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^{n}(I)$. …When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in C^{\infty}(I)$. …
###### Infinite Integrals
With $a, the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is …where the supremum is over all sets of points $x_{0} in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …
##### 10: Bibliography I
• IEEE (2015) IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..
• IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..
• Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_{0}(z)-iJ_{1}(z)$ and of Bessel functions $J_{m}(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.
• M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.