.ued在线最靠谱的博彩平台『世界杯佣金分红55%，咨询专员：@ky975』.ic6.k2q1w9-y0aqm00ek

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1: 36.5 Stokes Sets

… ►where $u$ satisfies the equation … ►Here $u$ is the root of the equation … ►where $u$ is the root of the equation …such that $u>\frac{1}{3}$. … ►where $u$ is the positive root of the equation …

2: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.13 $$\mathrm{Arcsin}u\pm \mathrm{Arcsin}v=\mathrm{Arcsin}\left(u{(1-v^2)}^{1/2}\pm v{(1-u^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.32

Permalink:

http://dlmf.nist.gov/4.24.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.15 $$\mathrm{Arctan}u\pm \mathrm{Arctan}v=\mathrm{Arctan}\left(\frac{u\pm v}{1\mp uv}\right),$$

ⓘ

Symbols:

$\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.34

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

ⓘ

Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

…

3: 12.6 Continued Fraction

… ►For a continued-fraction expansion of the ratio $U(a,x)/U(a-1,x)$ see Cuyt et al. (2008, pp. 340–341).

4: 12.4 Power-Series Expansions

… ►

12.4.1 $$U(a,z)=U(a,0)u_1(a,z)+U^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by ►

12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

…

5: 22.8 Addition Theorems

… ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►

22.8.14 $$\mathrm{sn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}u\mathrm{dn}v+\mathrm{sn}v\mathrm{cn}v\mathrm{dn}u}{\mathrm{cn}u\mathrm{cn}v+\mathrm{sn}u\mathrm{dn}u\mathrm{sn}v\mathrm{dn}v},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(ii), §22.8 and Ch.22

►

22.8.15 $$\mathrm{cn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}u\mathrm{dn}v-\mathrm{sn}v\mathrm{cn}v\mathrm{dn}u}{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}v-\mathrm{sn}v\mathrm{cn}u\mathrm{dn}u},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(ii), §22.8 and Ch.22

… ►

22.8.17 $$\mathrm{dn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}u-\mathrm{sn}v\mathrm{cn}u\mathrm{dn}v}{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}v-\mathrm{sn}v\mathrm{cn}u\mathrm{dn}u},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(ii), §22.8 and Ch.22

…

6: 3.4 Differentiation

… ►For partial derivatives we use the notation $u_{t,s}=u(x_0+th,y_0+sh)$. … ►

3.4.29 $${\nabla }^2{u}_{0,0}=\frac{1}{12h^2}\left(-60u_{0,0}+16(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1})-(u_{2,0}+u_{0,2}+u_{-2,0}+u_{0,-2})\right)+O\left(h^4\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.31

Permalink:

http://dlmf.nist.gov/3.4.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.34 $${\nabla }^4{u}_{0,0}=\frac{1}{6h^4}(184u_{0,0}-(u_{0,3}+u_{0,-3}+u_{3,0}+u_{-3,0})+14(u_{0,2}+u_{0,-2}+u_{2,0}+u_{-2,0})-77(u_{0,1}+u_{0,-1}+u_{1,0}+u_{-1,0})+20(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})-(u_{1,2}+u_{2,1}+u_{1,-2}+u_{2,-1}+u_{-1,2}+u_{-2,1}+u_{-1,-2}+u_{-2,-1}))+O\left(h^4\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.33

Permalink:

http://dlmf.nist.gov/3.4.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►For additional formulas involving values of ${\nabla }^{2}u$ and ${\nabla }^{4}u$ on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

7: 12.3 Graphics

… ►

►►

Figure 12.3.1: $U(a,x)$, $a$ = 0. … Magnify

… ►

►►

Figure 12.3.3: $U(a,x)$, $a$ = $-0.5$, $-2$, $-3.5$, $-5$. Magnify

… ►

►►

Figure 12.3.5: $U(-8,x)$, $\overline{U}(-8,x)$, $F(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

►

►►

Figure 12.3.6: $U^{\prime }(-8,x)$, ${\overline{U}}^{\prime }(-8,x)$, $G(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

►

►

►

Figure 12.3.7: $U(a,x)$, $-2.5\le a\le 2.5$, $-2.5\le x\le 2.5$. Magnify 3D Help

…

8: 32.13 Reductions of Partial Differential Equations

… ►The Korteweg–de Vries (KdV) equation … ►The sine-Gordon equation … ►

$u(x,t)=v(z),$

… ►The Boussinesq equation … ►

$u(x,t)=v(z),$

…

9: 36.9 Integral Identities

… ►

36.9.3 $${|{\mathrm{\Psi }}_1\left(x\right)|}^2=\sqrt{\frac{8\pi }{3}}{\int }_0^{\mathrm{\infty }}{u}^{-1/2}\cos \left(2u(x+u^2)+\frac{1}{4}\pi \right)du.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.4 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2={\int }_0^{\mathrm{\infty }}\left({\mathrm{\Psi }}_1\left(\frac{4u^3+2uy+x}{u^{1/3}}\right)+{\mathrm{\Psi }}_1\left(\frac{4u^3+2uy-x}{u^{1/3}}\right)\right)\frac{du}{u^{1/3}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.5 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2=2{\int }_0^{\mathrm{\infty }}\cos \left(2xu\right){\mathrm{\Psi }}_1\left(2u^{2/3}(y+2u^2)\right)\frac{du}{u^{1/3}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.6 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=2^{4/5}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\Psi }}_3\left(2^{4/5}(x+2uy+3u^{2}z+5u^4),0,2^{2/5}(z+10u^2)\right)du.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.7 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=\frac{2^{7/4}}{5^{1/4}}{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\left({\mathrm{e}}^{2\mathrm{i}u(u^4+z{u}^2+x)}{\mathrm{\Psi }}_2\left(\frac{2^{7/4}}{5^{1/4}}y{u}^{3/4},\sqrt{\frac{2u}{5}}(3z+10u^2)\right)\right)\frac{du}{u^{1/4}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

…

10: 21.9 Integrable Equations

… ►

21.9.1 $$4u_t=6u{u}_x+u_{xxx},$$

ⓘ

Symbols:

$u(x,y,t)$: elevation

Permalink:

http://dlmf.nist.gov/21.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

… ►

21.9.2 $$\mathrm{i}{u}_t=-\frac{1}{2}{u}_{xx}\pm {|u|}^{2}u.$$

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit and $u(x,y,t)$: elevation

Permalink:

http://dlmf.nist.gov/21.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

… ►

21.9.3 $${(-4u_t+6u{u}_x+u_{xxx})}_x+3u_{yy}=0.$$

ⓘ

Symbols:

$u(x,y,t)$: elevation

Referenced by:

Figure 21.9.2, Figure 21.9.2, §21.9, Sidebar 21.SB2, Sidebar 21.SB2

Permalink:

http://dlmf.nist.gov/21.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

►Here $x$ and $y$ are spatial variables, $t$ is time, and $u(x,y,t)$ is the elevation of the surface wave. …These parameters, including $𝛀$, are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition $u(x,y,0)$ (Deconinck and Segur (1998)). …