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31: 22.18 Mathematical Applications

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22.18.1 $$\left(x^2/a^2\right)+\left(y^2/b^2\right)=1,$$

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Symbols:

$x$: real, $y$: real, $a$: real and $b$: real

Permalink:

http://dlmf.nist.gov/22.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

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22.18.3 $$l(u)=aℰ(u,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $k$: modulus, $a$: real, $u$: variable and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

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22.18.7 $$a{x}^2{y}^2+b(x^{2}y+x{y}^2)+c(x^2+y^2)+2dxy+e(x+y)+f=0,$$

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Symbols:

$x$: real, $y$: real, $c$: real constant, $d$: real constant, $e$: real constant, $f$: real constant, $a$: real constant and $b$: real constant

Permalink:

http://dlmf.nist.gov/22.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(iii), §22.18 and Ch.22

►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …

32: 32.3 Graphics

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32.3.1 $$w_k(x)\sim k\mathrm{Ai}\left(x\right),$$ $x\to +\mathrm{\infty }$;

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $x$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(ii), §32.3 and Ch.32

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32.3.3 $$u\sim kU(-\nu -\frac{1}{2},x),$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $U(a,z)$: parabolic cylinder function, $x$: real, $k$: real, $\nu$: parameter and $u_k(x;\nu )$: solution

Permalink:

http://dlmf.nist.gov/32.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(iii), §32.3 and Ch.32

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32.3.4 $$w(x)=2\sqrt{2}{u}_k^2(\sqrt{2}x,\nu ),$$

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Symbols:

$x$: real, $k$: real, $\nu$: parameter and $u_k(x;\nu )$: solution

Permalink:

http://dlmf.nist.gov/32.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(iii), §32.3 and Ch.32

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32.3.5 $$w(x)\sim 2\sqrt{2}{k}^2{U}^2(-\nu -\frac{1}{2},\sqrt{2}x),$$ $x\to +\mathrm{\infty }$;

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Symbols:

$\sim$: asymptotic equality, $U(a,z)$: parabolic cylinder function, $x$: real, $k$: real and $\nu$: parameter

Permalink:

http://dlmf.nist.gov/32.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(iii), §32.3 and Ch.32

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32.3.6 $$u^2=-\frac{1}{3}x\pm \frac{1}{6}\sqrt{x^2+12\nu +6}.$$

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Symbols:

$x$: real, $\nu$: parameter and $u_k(x;\nu )$: solution

Permalink:

http://dlmf.nist.gov/32.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(iii), §32.3 and Ch.32

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33: 10.76 Approximations

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Real Variable and Order $:$ Functions

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Real Variable and Order $:$ Zeros

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Real Variable and Order $:$ Integrals

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Complex Variable; Real Order

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Real Variable; Imaginary Order

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34: 5.3 Graphics

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§5.3(i) Real Argument

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35: 4.1 Special Notation

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$k,m,n$	integers.
$a,c$	real or complex constants.
$x,y$	real variables.
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►It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments $x$. …

36: 9.20 Software

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§9.20(ii) $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, $x\in ℝ$

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§9.20(iv) Real and Complex Zeros

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§9.20(v) Integrals of $\mathrm{Ai}\left(x\right)$, $\mathrm{Bi}\left(x\right)$, $x\in ℝ$

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37: Software Index

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	Open Source													With Book						Commercial
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16.27(ii) Real Arguments	✓		✓		✓	✓						a	✓								✓	✓	✓
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20.16(ii) Real arguments	✓											a	✓	✓			✓				✓	✓	a	✓
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22.22(ii) Real Argument	✓		✓		✓	✓		✓				✓	✓	✓		✓	✓		✓	✓	✓	✓	✓	✓
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23.24(ii) Real Argument																					✓	✓	a
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33.26(ii) Real arguments	✓				✓				✓			a	✓	✓			✓				✓		a
…

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38: 29.4 Graphics

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39: 36.4 Bifurcation Sets

… ►These are real solutions $t_j(𝐱)$, $1\le j\le j_{\max }(𝐱)\le K+1$, of … ►These are real solutions $\{s_j(𝐱),t_j(𝐱)\}$, $1\le j\le j_{\max }(𝐱)\le 4$, of … ►

36.4.5 $$x=0.$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.6 $$27x^2=-8y^3.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.13 $$x=y=-\frac{1}{4}{z}^2.$$

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Symbols:

$y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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40: 12.12 Integrals

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12.12.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{\mu -1}U(a,t)dt=\frac{\sqrt{\pi }{2}^{-\frac{1}{2}(\mu +a+\frac{1}{2})}\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}(\mu +a+\frac{3}{2})\right)},$$ $\mathrm{\Re }\mu >0$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{3}{4}{t}^2}{t}^{-a-\frac{3}{2}}U(a,t)dt=2^{\frac{1}{4}+\frac{1}{2}a}\mathrm{\Gamma }\left(-a-\frac{1}{2}\right)\cos \left((\frac{1}{4}a+\frac{1}{8})\pi \right),$$ ,

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{-a-\frac{1}{2}}{(x^2+t^2)}^{-1}U(a,t)dt=\sqrt{\pi /2}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)x^{-a-\frac{3}{2}}{\mathrm{e}}^{\frac{1}{4}{x}^2}U(-a,x),$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $x$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

►When $z$ $(=x)$ is real the left-hand side equals ${(F(a,x))}^2$; compare (12.2.22). …