elliptic cases

(0.004 seconds)

11—20 of 67 matching pages

11: 19.2 Definitions

… ►Also, if

k^{2}

and

\alpha^{2}

are real, then

\Pi\left(\phi,\alpha^{2},k\right)

is called a circular or hyperbolic case according as

\alpha^{2}(\alpha^{2}-k^{2})(\alpha^{2}-1)

is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points

\alpha^{2}=0,k^{2},1

. … ►Formulas involving

\Pi\left(\phi,\alpha^{2},k\right)

that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using

R_{C}\left(x,y\right)

. …

12: 19.5 Maclaurin and Related Expansions

… ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). …

13: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►There are four important special cases of (19.14.4)–(19.14.6), as follows. … ►(These four cases include 12 integrals in Abramowitz and Stegun (1964, p. 596).) ►

§19.14(ii) General Case

… ►

14: 19.36 Methods of Computation

… ►Also, see Todd (1975) for a special case of

K\left(k\right)

. …

15: 19.13 Integrals of Elliptic Integrals

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for

F\left(\phi,k\right)

and

E\left(\phi,k\right)

, together with special cases. …

16: 22.19 Physical Applications

… ►

Case I: $V(x)=\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}$

… ►A more abstract overview is Audin (1999, Chapters III and IV), and a complete discussion of analytical solutions in the elliptic and hyperelliptic cases appears in Golubev (1960, Chapters V and VII), the original hyperelliptic investigation being due to Kowalevski (1889). …

17: 22.20 Methods of Computation

§22.20 Methods of Computation

… ►

§22.20(iii) Landen Transformations

… ►

§22.20(iv) Lattice Calculations

… ►

§22.20(v) Inverse Functions

… ►

18: 19.33 Triaxial Ellipsoids

… ►A conducting elliptic disk is included as the case

c=0

. …

19: 22.10 Maclaurin Series

… ►

§22.10(i) Maclaurin Series in $z$

… ►The full expansions converge when

|z|<\min\left(K\left(k\right),{K^{\prime}}\left(k\right)\right)

. ►

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$

… ►The radius of convergence is the distance to the origin from the nearest pole in the complex

k

-plane in the case of (22.10.4)–(22.10.6), or complex

k^{\prime}

-plane in the case of (22.10.7)–(22.10.9); see §22.17.

20: 20.9 Relations to Other Functions

… ►

§20.9(i) Elliptic Integrals

… ►In the case of the symmetric integrals, with the notation of §19.16(i) we have … ►

§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►As a function of

\tau

k^{2}

is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6). …