exponential%20integrals

SearchAdvancedHelp

(0.004 seconds)

21—30 of 37 matching pages

21: Bibliography S

… ►

Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

… ►

Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Algorithm 911: multiple-precision exponential integral and related functions. ACM Trans. Math. Software 37 (4), pp. Art. 46, 16.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

5th item, §4.48(iii), 3rd item, §6.21(ii).

Encodings:

BibTeX

… ►

Automatic computing methods for special functions. II. The exponential integral $E_n(x)$ . J. Res. Nat. Bur. Standards Sect. B 78B, pp. 199–216.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 16S.

External Links:

MathReview (Bernard H. Rosman), ZentralBlatt

Referenced by:

§8.25(v), §8.28(vi).

Encodings:

BibTeX

►

Automatic computing methods for special functions. III. The sine, cosine, exponential integrals, and related functions. J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 291–311.

ⓘ

Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

MathReview (Martin E. Price), ZentralBlatt

Referenced by:

§6.21(ii).

Encodings:

BibTeX

…

22: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►

12.10.3 $$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{g(\mu ){\mathrm{e}}^{-{\mu }^2\xi }}{{(t^2-1)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}\frac{{𝒜}_s(t)}{{\mu }^{2s}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$ , ${𝒜}_s(t)$: coefficients and $g(\mu )$: expansion

Referenced by:

§12.10(ii), §12.10(iii), §12.10(vi), §12.10(vi)

Permalink:

http://dlmf.nist.gov/12.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(ii), §12.10 and Ch.12

►

12.10.4 $$U^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim -\frac{\mu }{\sqrt{2}}g(\mu ){(t^2-1)}^{\frac{1}{4}}{\mathrm{e}}^{-{\mu }^2\xi }\sum _{s=0}^{\mathrm{\infty }}\frac{{ℬ}_s(t)}{{\mu }^{2s}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\xi$ , ${ℬ}_s(t)$: coefficients and $g(\mu )$: expansion

Referenced by:

§12.10(ii)

Permalink:

http://dlmf.nist.gov/12.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(ii), §12.10 and Ch.12

… ►

12.10.15 $$h(\mu )=2^{-\frac{1}{4}{\mu }^2-\frac{1}{4}}{\mathrm{e}}^{-\frac{1}{4}{\mu }^2}{\mu }^{\frac{1}{2}{\mu }^2-\frac{1}{2}},$$

ⓘ

Defines:

$h(\mu )$: expansion (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm

Referenced by:

§12.10(vi)

Permalink:

http://dlmf.nist.gov/12.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(ii), §12.10 and Ch.12

… ►

12.10.33 $${𝖠}_{s+1}(\tau )=-4{\tau }^2{(\tau +1)}^2\frac{d}{d\tau }{𝖠}_s(\tau )-\frac{1}{4}{\int }_0^{\tau }\left(20u^2+20u+3\right){𝖠}_s(u)du,$$ $s=0,1,2,\mathrm{\dots }$,

ⓘ

Defines:

${𝖠}_s(\tau )$: coefficients (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, $s$: nonnegative integer and $\tau$

Permalink:

http://dlmf.nist.gov/12.10.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vi), §12.10 and Ch.12

… ►

$${𝖠}_1(\tau )=-\frac{1}{12}\tau (20{\tau }^2+30\tau +9),$$

…

23: 36.4 Bifurcation Sets

… ► $K=1$, fold bifurcation set: … ►

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

… ►

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

… ►

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

►

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

…

24: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
6 Exponential, Logarithmic, Sine, and Cosine Integrals
6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓		✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$	✓	✓				✓			✓			✓	✓	✓							✓	✓	✓
…
8.28(vii) $E_p\left(z\right)$, $z,p\in ℂ$	✓											✓	✓								✓	✓	✓
…
20 Theta Functions
…

…

25: 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 … ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). …

26: 3.4 Differentiation

… ►If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ► $f(z)={\mathrm{e}}^z$, $x_0=0$. The integral (3.4.18) becomes …With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. …

27: Bibliography B

… ►

Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

… ►

A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

… ►

Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

… ►

Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

…

28: Bibliography C

… ►

A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.

ⓘ

External Links:

ISSN 0898-1221, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

►

On the evaluation of generalized exponential integrals $E_v(x)$ . J. Comput. Phys. 78 (2), pp. 278–287.

ⓘ

External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§8.25(v), 2nd item.

Encodings:

BibTeX

►

An algorithm for exponential integrals of real order. Computing 45 (3), pp. 269–276.

ⓘ

Notes:

Includes Fortran code

External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§8.25(v), §8.28(vi).

Encodings:

BibTeX

►

On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

ⓘ

Notes:

Includes Fortran code

External Links:

Document, ZentralBlatt

Referenced by:

§8.28(vi).

Encodings:

BibTeX

… ►

Applications of the complex exponential integral. Math. Comp. 15 (73), pp. 1–6.

ⓘ

External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§6.7(iv).

Encodings:

BibTeX

…

29: 11.6 Asymptotic Expansions

… ►

11.6.3 $${\int }_0^z{𝐊}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.32

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

►

11.6.4 $${\int }_0^z{𝐌}_0\left(t\right)dt+\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.8

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

… ►

11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

… ►

$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

…

30: 2.11 Remainder Terms; Stokes Phenomenon

… ►From §8.19(i) the generalized exponential integral is given by …However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …