Alanus-Universit?t证书办理【购证 微kaa77788】4YVBYAE

SearchAdvancedHelp

(0.023 seconds)

21—30 of 517 matching pages

21: Ranjan Roy

… ►

4 Elementary Functions

… ►In November 2015, Roy was named Associate Editor for Chapters 1 and 4.

22: 12.9 Asymptotic Expansions for Large Variable

… ►

12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

…

23: 18.4 Graphics

… ►

►►

Figure 18.4.1: Jacobi polynomials $P_n^{(1.5,-0.5)}\left(x\right)$, $n=1,2,3,4,5$. Magnify

… ►

►►

Figure 18.4.3: Chebyshev polynomials $T_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

►

►►

Figure 18.4.4: Legendre polynomials $P_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

►

►►

Figure 18.4.5: Laguerre polynomials $L_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

►

►►

Figure 18.4.6: Laguerre polynomials $L_3^{(\alpha )}\left(x\right)$, $\alpha =0,1,2,3,4$. Magnify

…

24: 30.16 Methods of Computation

… ►

$$A_{j,j+1}=-{\gamma }^2\frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)},$$

… ►For $m=2$, $n=4$, ${\gamma }^2=10$, … ►

${\alpha }_{2,4}=\mathrm{13.97907\hspace{0.33em}459},$

… ►which yields ${\lambda }_4^2\left(10\right)=\mathrm{13.97907\hspace{0.33em}345}$. … ►The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11). …

25: 4.9 Continued Fractions

… ►

4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

►

4.9.2 $$\ln \left(\frac{1+z}{1-z}\right)=\frac{2z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\frac{16z^2}{9-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.40

Permalink:

http://dlmf.nist.gov/4.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

… ►

$$=1+\frac{z}{1-(z/2)+}\frac{z^2/(4\cdot 3)}{1+}\frac{z^2/(4\cdot 15)}{1+}\frac{z^2/(4\cdot 35)}{1+}\mathrm{\cdots }\frac{z^2/(4(4n^2-1))}{1+}\mathrm{\cdots }$$

►

4.9.4 $${\mathrm{e}}^z-e_{n-1}(z)=\frac{z^n}{n!-}\frac{n!z}{(n+1)+}\frac{z}{(n+2)-}\frac{(n+1)z}{(n+3)+}\frac{2z}{(n+4)-}\frac{(n+2)z}{(n+5)+}\frac{3z}{(n+6)-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $z$: complex variable and $e_n(z)$: expansion of exponential

A&S Ref:

4.2.41

Permalink:

http://dlmf.nist.gov/4.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(ii), §4.9 and Ch.4

…

26: 10.65 Power Series

… ►

${\mathrm{ber}}_{\nu }x={(\frac{1}{2}x)}^{\nu }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(\frac{3}{4}\nu \pi +\frac{1}{2}k\pi \right)}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}}{(\frac{1}{4}{x}^2)}^k,$

►

${\mathrm{bei}}_{\nu }x={(\frac{1}{2}x)}^{\nu }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left(\frac{3}{4}\nu \pi +\frac{1}{2}k\pi \right)}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}}{(\frac{1}{4}{x}^2)}^k.$

►

$\mathrm{ber}x=1-{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^2}{{(2!)}^2}}+{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^4}{{(4!)}^2}}-\mathrm{\cdots },$

►

$\mathrm{bei}x=\frac{1}{4}{x}^2-{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^3}{{(3!)}^2}}+{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^5}{{(5!)}^2}}-\mathrm{\cdots }.$

… ►

$$\ker x=-\ln \left(\frac{1}{2}x\right)\mathrm{ber}x+\frac{1}{4}\pi \mathrm{bei}x+\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(2k+1\right)}{{((2k)!)}^2}{(\frac{1}{4}{x}^2)}^{2k},$$

…

27: 12.14 The Function $W(a,x)$

… ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►

12.14.13 $$W(0,\pm x)=2^{-\frac{5}{4}}\sqrt{\pi x}\left(J_{-\frac{1}{4}}\left(\frac{1}{4}{x}^2\right)\mp J_{\frac{1}{4}}\left(\frac{1}{4}{x}^2\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $W(a,x)$: parabolic cylinder function and $x$: real variable

A&S Ref:

19.25.4 (corrected)

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/12.14.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(vii), §12.14(vii), §12.14 and Ch.12

►

12.14.14 $$\frac{d}{dx}W(0,\pm x)=-2^{-\frac{9}{4}}x\sqrt{\pi x}\left(J_{\frac{3}{4}}\left(\frac{1}{4}{x}^2\right)\pm J_{-\frac{3}{4}}\left(\frac{1}{4}{x}^2\right)\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $W(a,x)$: parabolic cylinder function and $x$: real variable

A&S Ref:

19.25.5

Permalink:

http://dlmf.nist.gov/12.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(vii), §12.14(vii), §12.14 and Ch.12

… ►

12.14.20 $$s_1(a,x)\sim 1+\frac{d_2}{1!2x^2}-\frac{c_4}{2!2^2{x}^4}-\frac{d_6}{3!2^3{x}^6}+\frac{c_8}{4!2^4{x}^8}+\mathrm{\cdots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $x$: real variable, $a$: real or complex parameter, $c_{2r}$: coefficients, $d_{2r}$: coefficients and $s_n(a,x)$: coefficients

A&S Ref:

19.21.5

Permalink:

http://dlmf.nist.gov/12.14.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(viii), §12.14 and Ch.12

… ►

12.14.29 $$l(\mu )\sim \frac{2^{\frac{1}{4}}}{{\mu }^{\frac{1}{2}}}\sum _{s=0}^{\mathrm{\infty }}\frac{l_s}{{\mu }^{4s}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $l(\mu )$ and $l_s$: coefficient

Permalink:

http://dlmf.nist.gov/12.14.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

…

28: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. … ►with diffusion constant $\alpha =\pi /4$. … ►

20.13.3 $$g(z,t)=\sqrt{\frac{\pi }{4\alpha t}}\exp \left(-\frac{z^2}{4\alpha t}\right)$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $t$: time, $\alpha$: real parameter and $z$: real variable

Permalink:

http://dlmf.nist.gov/20.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

29: 10.3 Graphics

… ►

►

►

Figure 10.3.9: $J_0\left(x+\mathrm{i}y\right)$, $-10\le x\le 10$, $-4\le y\le 4$. Magnify 3D Help

… ►

►

►

Figure 10.3.11: $J_1\left(x+\mathrm{i}y\right)$, $-10\le x\le 10$, $-4\le y\le 4$. Magnify 3D Help

… ►

►

►

Figure 10.3.13: $J_5\left(x+\mathrm{i}y\right)$, $-10\le x\le 10$, $-4\le y\le 4$. Magnify 3D Help

►

►

►

Figure 10.3.14: $H_5^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

►

►

►

Figure 10.3.15: $J_{5.5}\left(x+\mathrm{i}y\right)$, $-10\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

…

30: 16.14 Partial Differential Equations

… ►

$$x(1-x)\frac{{\partial }^2{F}_4}{{\partial x}^2}-2xy\frac{{\partial }^2{F}_4}{\partial x\partial y}-y^2\frac{{\partial }^2{F}_4}{{\partial y}^2}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_4}{\partial x}-(\alpha +\beta +1)y\frac{\partial F_4}{\partial y}-\alpha \beta F_4=0,$$

►

$$y(1-y)\frac{{\partial }^2{F}_4}{{\partial y}^2}-2xy\frac{{\partial }^2{F}_4}{\partial x\partial y}-x^2\frac{{\partial }^2{F}_4}{{\partial x}^2}+\left({\gamma }^{\prime }-(\alpha +\beta +1)y\right)\frac{\partial F_4}{\partial y}-(\alpha +\beta +1)x\frac{\partial F_4}{\partial x}-\alpha \beta F_4=0.$$

… ►

16.14.6 $$G_3(\alpha ,{\alpha }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\alpha +2n-m\right)\mathrm{\Gamma }\left({\alpha }^{\prime }+2m-n\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left({\alpha }^{\prime }\right)}\frac{x^m{y}^n}{m!n!},$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.14(ii), §16.14 and Ch.16

►(The region of convergence is not quite maximal.) …