exponential%20growth

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21: Daniel W. Lozier

… ►Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. …

22: 36.4 Bifurcation Sets

… ►

$x=\frac{9}{20}{z}^2.$

… ►

$x=-\frac{3}{20}{z}^2,$

… ►

36.4.11 $$x+\mathrm{i}y=-z^2\exp \left(\frac{2}{3}\mathrm{i}\pi m\right),$$ $m=0,1,2$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $n$: integer and $x$: real parameter

Referenced by:

§36.7(iii)

Permalink:

http://dlmf.nist.gov/36.4.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

$x=-\frac{1}{12}{z}^2(\exp \left(2\tau \right)\pm 2\exp \left(-\tau \right)),$

►

$y=-\frac{1}{12}{z}^2(\exp \left(-2\tau \right)\pm 2\exp \left(\tau \right))$ ,

…

23: 1.14 Integral Transforms

… ►

1.14.3 $$\frac{1}{2}(f(u+)+f(u-))=\frac{1}{\sqrt{2\pi }}{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}F(x){\mathrm{e}}^{-\mathrm{i}xu}dx,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value and $F(x)$: Fourier transform of $f(t)$

Permalink:

http://dlmf.nist.gov/1.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

… ►Assume that $f(t)$ is piecewise continuous on $[0,\mathrm{\infty })$ and of exponential growth, that is, constants $M$ and $\alpha$ exist such that … ►where $f_a(t)={\mathrm{e}}^{at}f(t)$. … ►

1.14.25 $$ℒf\left(s\right)=\frac{1}{1-{\mathrm{e}}^{-as}}{\int }_0^a{\mathrm{e}}^{-st}f(t)dt.$$

ⓘ

Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(s\right)$ from $ℒ(f(t);s)$.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►

1.14.26 $$ℒf\left(s\right)=\frac{1}{1+{\mathrm{e}}^{-as}}{\int }_0^a{\mathrm{e}}^{-st}f(t)dt.$$

ⓘ

Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E26

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(s\right)$ from $ℒ(f(t);s)$.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

…

24: 36.2 Catastrophes and Canonical Integrals

… ►

36.2.6 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2\sqrt{\pi /3}\exp \left(\mathrm{i}\left(\frac{4}{27}{z}^3+\frac{1}{3}xz-\frac{1}{4}\pi \right)\right){\int }_{\mathrm{\infty }\exp \left(-7\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(u^6+2z{u}^4+(z^2+x)u^2+\frac{y^2}{12u^2}\right)\right)du,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii), §36.5(iii)

Permalink:

http://dlmf.nist.gov/36.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)={\displaystyle \frac{4\pi }{3^{1/3}}}\exp \left(\mathrm{i}\left(\frac{2}{27}{z}^3-\frac{1}{3}xz\right)\right)\left(\exp \left(-\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{+}(𝐱)+\exp \left(\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{-}(𝐱)\right),$

… ►

36.2.12 $${\mathrm{\Psi }}_0=\sqrt{\pi }\exp \left(\mathrm{i}\frac{\pi }{4}\right).$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function and $\mathrm{i}$: imaginary unit

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

… ►

36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

►

36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

ⓘ

Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

25: 28.29 Definitions and Basic Properties

… ►

28.29.7 $$w(z+\pi )={\mathrm{e}}^{\pi \mathrm{i}\nu }w(z),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: complex parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

►iff ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ is an eigenvalue of the matrix … ►

28.29.10 $$F_{\nu }(z)={\mathrm{e}}^{\mathrm{i}\nu z}{P}_{\nu }(z),$$

ⓘ

Defines:

$F_{\nu }(z)$: Floquet solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.29.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►A nontrivial solution $w(z)$ is either a Floquet solution with respect to $\nu$, or $w(z+\pi )-{\mathrm{e}}^{\mathrm{i}\nu \pi }w(z)$ is a Floquet solution with respect to $-\nu$. … ►Its order of growth for $|\lambda |\to \mathrm{\infty }$ is exactly $\frac{1}{2}$; see Magnus and Winkler (1966, Chapter II, pp. 19–28). …

26: 6.8 Inequalities

§6.8 Inequalities

… ►

6.8.1

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.20

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

►

6.8.2

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

►

6.8.3

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

27: 12.19 Tables

… ►

Karpov and Čistova (1968) includes ${\mathrm{e}}^{-\frac{1}{4}{x}^2}{D}_p\left(-x\right)$ and ${\mathrm{e}}^{-\frac{1}{4}{x}^2}{D}_p\left(\mathrm{i}x\right)$ for $x=0(.01)5$ and $x^{-1}$ = 0(.001 or .0001)5, $p=-1(.1)1$, 7D or 8S.

►

Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U(n-\frac{1}{2},x)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.

…

28: 6.5 Further Interrelations

§6.5 Further Interrelations

… ►

6.5.1 $$E_1\left(-x\pm \mathrm{i}0\right)=-\mathrm{Ei}\left(x\right)\mp \mathrm{i}\pi ,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

►

6.5.2 $$\mathrm{Ei}\left(x\right)=-\frac{1}{2}(E_1\left(-x+\mathrm{i}0\right)+E_1\left(-x-\mathrm{i}0\right)),$$

ⓘ

Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

►

6.5.3 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)+E_1\left(x\right))=\mathrm{Shi}\left(x\right)=-\mathrm{i}\mathrm{Si}\left(\mathrm{i}x\right),$$

ⓘ

Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Shi}\left(z\right)$: hyperbolic sine integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $x$: real variable

A&S Ref:

5.2.22 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

… ►

6.5.6 $$\mathrm{Ci}\left(z\right)=-\frac{1}{2}(E_1\left(\mathrm{i}z\right)+E_1\left(-\mathrm{i}z\right)),$$

ⓘ

Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

5.2.23

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

…

29: 20.11 Generalizations and Analogs

… ►

20.11.1 $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$

ⓘ

Defines:

$G(m,n)$: Gauss sum (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

… ►

20.11.2 $$\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}\sum _{j=0}^{m-1}{\mathrm{e}}^{\pi \mathrm{i}{j}^{2}n/m}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}G(-n,m).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $G(m,n)$: Gauss sum

Permalink:

http://dlmf.nist.gov/20.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

… ►With the substitutions $a=q{\mathrm{e}}^{2\mathrm{i}z}$, $b=q{\mathrm{e}}^{-2\mathrm{i}z}$, with $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, we have …

30: Bibliography H

… ►

Formulae for growth factors in expanding universes containing matter and a cosmological constant. Monthly Notices Roy. Astronom. Soc. 322 (2), pp. 419–425.

ⓘ

External Links:

Document

Referenced by:

§8.24(ii).

Encodings:

BibTeX

… ►

Spherical Bessel expansions of sine, cosine, and exponential integrals. Appl. Numer. Math. 34 (1), pp. 95–98.

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External Links:

ISSN 0168-9274, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii), §6.10(ii), §6.15.

Encodings:

BibTeX

… ►

Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.

ⓘ

External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

… ►

Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87 (3), pp. 591–609.

ⓘ

External Links:

ISSN 0024-6115, Document, Link, MathReview (Angel V. Kumchev), ZentralBlatt

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11.

Encodings:

BibTeX