DLMF: Untitled Document

… ►

W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

…

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36.2.28

\Psi^{(\mathrm{E})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{E})}\left(0,0,-% z\right)}\\ =2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6% }\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right),

z\geq 0

,

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

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

-\infty<z<\infty

.

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : 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

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	Open Source								With Book		Commercial
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19 Elliptic Integrals
19.39(ii) $K\left(k\right)$ , $E\left(k\right)$ , $0\leq k^{2}\leq 1$	✓	✓	✓	✓	✓	✓	✓	✓			✓	✓	✓	✓
19.39(iii) $F\left(\phi,k\right)$ , $E\left(\phi,k\right)$ , $\phi,k\in\mathbb{R}$	✓	✓	✓	✓	✓	✓	✓	✓		✓		✓	✓	a	✓
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20 Theta Functions
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22 Jacobian Elliptic Functions
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§19.36 Methods of Computation

… ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … ►

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

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23 Weierstrass Elliptic and Modular Functions

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§22.3(i) Real Variables: Line Graphs

►Line graphs of the functions

\operatorname{sn}\left(x,k\right)

,

\operatorname{cn}\left(x,k\right)

,

\operatorname{dn}\left(x,k\right)

,

\operatorname{cd}\left(x,k\right)

,

\operatorname{sd}\left(x,k\right)

,

\operatorname{nd}\left(x,k\right)

,

\operatorname{dc}\left(x,k\right)

,

\operatorname{nc}\left(x,k\right)

,

\operatorname{sc}\left(x,k\right)

,

\operatorname{ns}\left(x,k\right)

,

\operatorname{ds}\left(x,k\right)

, and

\operatorname{cs}\left(x,k\right)

for representative values of real

x

and real

k

illustrating the near trigonometric (

k=0

), and near hyperbolic (

k=1

) limits. … ►

\operatorname{sn}\left(x,k\right)

,

\operatorname{cn}\left(x,k\right)

, and

\operatorname{dn}\left(x,k\right)

as functions of real arguments

x

and

k

. … ►

§22.3(iii) Complex $z$ ; Real $k$

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§22.3(iv) Complex $k$

…

… ►

Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

{K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}

{E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}

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Equations (22.14.16), (22.14.17)

22.14.16

\int_{0}^{K\left(k\right)}\ln\left(\operatorname{sn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)-\tfrac{1}{2}K\left(k% \right)\ln k,

22.14.17

\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k% \right)\ln\left(k^{\prime}/k\right)

Originally, a factor of $\pi$ was missing from the terms containing the $-\tfrac{1}{4}{K^{\prime}}\left(k\right)$ .

Reported by Fred Hucht on 2020-08-06

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Subsection 19.16(iii)

Originally it was implied that $R_{C}\left(x,y\right)$ is an elliptic integral. It was clarified that $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, $R_{C}\left(x,y\right)$ does not satisfy these conditions.

Reported 2010-11-23.

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Equation (36.10.14)

36.10.14

3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2\mathrm{i}z\frac{% \partial\Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0

Originally this equation appeared with $\frac{\partial\Psi^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial\Psi^{(\mathrm{E})}}{\partial x}$ .

Reported 2010-04-02.

…

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H. A. Ragheb, L. Shafai, and M. Hamid (1991) Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.

ⓘ

External Links:: Document
Referenced by:: 6th item.
Encodings:: BibTeX

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H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:: MathReview (H. H. Martens), ZentralBlatt
Referenced by:: §20.11(iv), §20.12(ii).
Encodings:: BibTeX

►

J. Raynal (1979) On the definition and properties of generalized

6

-

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(i), §19.36(ii).
Encodings:: BibTeX

… ►

H. Rosengren (2004) Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §20.11(iii).
Encodings:: BibTeX

…

… ►

H. Alzer and S. Qiu (2004) Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen (2000) Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

ⓘ

External Links:: ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.35(i).
Encodings:: BibTeX

►

G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1990) Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

G. D. Anderson and M. K. Vamanamurthy (1985) Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.

ⓘ

External Links:: ISSN 0350-1302, MathReview, ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

…

elliptic%20integrals

1—10 of 27 matching pages

1: Bibliography N

2: 36.2 Catastrophes and Canonical Integrals

3: Software Index

4: 19.36 Methods of Computation

§19.36 Methods of Computation

5: Bibliography F

6: Peter L. Walker

7: 22.3 Graphics

§22.3(i) Real Variables: Line Graphs

§22.3(iii) Complex $z$ ; Real $k$

§22.3(iv) Complex $k$

8: Errata

9: Bibliography R

10: Bibliography

elliptic%20integrals

1—10 of 27 matching pages

1: Bibliography N

2: 36.2 Catastrophes and Canonical Integrals

3: Software Index

4: 19.36 Methods of Computation

§19.36 Methods of Computation

5: Bibliography F

6: Peter L. Walker

7: 22.3 Graphics

§22.3(i) Real Variables: Line Graphs

§22.3(iii) Complex z ; Real k

§22.3(iv) Complex k

8: Errata

9: Bibliography R

10: Bibliography

§22.3(iii) Complex $z$ ; Real $k$

§22.3(iv) Complex $k$