along%20real%20line

SearchAdvancedHelp

(0.002 seconds)

21—30 of 684 matching pages

21: 6.3 Graphics

… ►

§6.3(i) Real Variable

… ►

►

►

Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. …There is a cut along the negative real axis. … Magnify 3D Help

22: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. … ►For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu$. … ►Also, all nontrivial solutions of (28.2.1) are unbounded on $ℝ$. ►For real $a$ and $q$ $(\ne 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. …

23: 10.42 Zeros

… ►For example, if $\nu$ is real, then the zeros of $I_{\nu }\left(z\right)$ are all complex unless for some positive integer $\mathrm{\ell }$, in which event $I_{\nu }\left(z\right)$ has two real zeros. ►The distribution of the zeros of $K_n\left(nz\right)$ in the sector $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi$ in the cases $n=1,5,10$ is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle $-\frac{1}{2}\pi$ so that in each case the cut lies along the positive imaginary axis. … ► $K_n\left(z\right)$ has no zeros in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi$; this result remains true when $n$ is replaced by any real number $\nu$. For the number of zeros of $K_{\nu }\left(z\right)$ in the sector $|\mathrm{ph}z|\le \pi$, when $\nu$ is real, see Watson (1944, pp. 511–513). …

24: 11.3 Graphics

… ►

►

►

Figure 11.3.8: $\left|{𝐊}_0\left(x+\mathrm{i}y\right)\right|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

►

►

►

Figure 11.3.9: $|{𝐇}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

►

►

►

Figure 11.3.10: $|{𝐊}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

… ►

►

►

Figure 11.3.12: $\left|{𝐊}_1\left(x+\mathrm{i}y\right)\right|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

… ►

►

►

Figure 11.3.19: $|{𝐌}_{-\frac{1}{2}}\left(x+\mathrm{i}y\right)|$ (principal value) for $-3\le x\le 3$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

…

25: 8.26 Tables

… ►

Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

… ►

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

26: 23 Weierstrass Elliptic and Modular
Functions

…

27: 13.4 Integral Representations

… ►

§13.4(i) Integrals Along the Real Line

… ►where $c$ is arbitrary, $\mathrm{\Re }c>0$. … ►The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$. …The contour cuts the real axis between $-1$ and $0$. … ►Again, $t^{-c}$ and the ${}_2{}^{}𝐅_1^{}$ function assume their principal values where the contour intersects the positive real axis. …

28: 6.19 Tables

… ►

§6.19(ii) Real Variables

… ►

Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

… ►

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 6D; $E_1\left(z\right)+\ln z$, $x=-2(.5)2.5$, $y=0(.2)1$, 6D.

►

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_1\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.

29: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. … ►

20 Theta Functions

…

30: 10.32 Integral Representations

… ►

§10.32(i) Integrals along the Real Line

… ►

10.32.7 $$K_{\nu }\left(x\right)=\sec \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\cos \left(x\sinh t\right)\cosh \left(\nu t\right)dt=\csc \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\sin \left(x\sinh t\right)\sinh \left(\nu t\right)dt,$$ , $x>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.6.22

Referenced by:

§10.32(i)

Permalink:

http://dlmf.nist.gov/10.32.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

… ►

10.32.11 $$K_{\nu }\left(xz\right)=\frac{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right){(2z)}^{\nu }}{{\pi }^{\frac{1}{2}}{x}^{\nu }}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(xt\right)dt}{{(t^2+z^2)}^{\nu +\frac{1}{2}}},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$, $x>0$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.25 (corrected)

Permalink:

http://dlmf.nist.gov/10.32.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32(i), §10.32 and Ch.10

… ►

10.32.15 $$I_{\mu }\left(z\right)I_{\nu }\left(z\right)=\frac{2}{\pi }{\int }_0^{\frac{1}{2}\pi }{I}_{\mu +\nu }\left(2z\cos \theta \right)\cos \left((\mu -\nu )\theta \right)d\theta ,$$ $\mathrm{\Re }\left(\mu +\nu \right)>-1$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.32.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32 and Ch.10

►

10.32.16 $$I_{\mu }\left(x\right)K_{\nu }\left(x\right)={\int }_0^{\mathrm{\infty }}{J}_{\mu \pm \nu }\left(2x\sinh t\right){\mathrm{e}}^{(-\mu \pm \nu )t}dt,$$ $\mathrm{\Re }\left(\mu \mp \nu \right)>-\frac{1}{2}$, $\mathrm{\Re }\left(\mu \pm \nu \right)>-1$, $x>0$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter

Keywords:

Laplace transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32 and Ch.10

…