of squares

SearchAdvancedHelp

(0.001 seconds)

31—40 of 87 matching pages

31: 23.22 Methods of Computation

… ►

In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

ⓘ

Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

►

If $d=0$, then

23.22.2 $$2{\omega }_1=-2\mathrm{i}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{2\sqrt{\pi }{c}^{1/4}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§22.20(iv), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 4 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and ${\omega }_1>0$.

…

32: 20.7 Identities

… ►

§20.7(i) Sums of Squares

… ►In the following equations ${\tau }^{\prime }=-1/\tau$, and all square roots assume their principal values. …

33: 1.4 Calculus of One Variable

… ►

Square-Integrable Functions

►A function $f(x)$ is square-integrable if …

34: 19.21 Connection Formulas

… ►

19.21.7 $$(x-y)R_D(y,z,x)+(z-y)R_D(x,y,z)=3R_F(x,y,z)-3y^{1/2}{x}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.10 $$2R_G(x,y,z)=z{R}_F(x,y,z)-\frac{1}{3}(x-z)(y-z)R_D(x,y,z)+x^{1/2}{y}^{1/2}{z}^{-1/2},$$ $z\ne 0$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)

Permalink:

http://dlmf.nist.gov/19.21.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

An incorrect factor of 3 was removed in the last term on the right-hand side.

Suggested 2022-06-27 by Abdulhafeez Abdulsalam

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

…

35: Guide to Searching the DLMF

… ►

Table 1: Query Examples

► ►►►

Query	Matching records contain
…
trigonometric^2 + trig$^2	any sum of the squares of two trigonometric functions such as ${\sin }^{2}z+{\cos }^{2}z$.
…

►

…

36: 4.16 Elementary Properties

… ►

Table 4.16.3: Trigonometric functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

► ►►

	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
…

►

37: 19.34 Mutual Inductance of Coaxial Circles

… ►is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …

38: 22.16 Related Functions

… ►See Figure 22.16.2. … ►

22.16.26 $$ℰ(x,k)=-{\int }_0^x\left({\mathrm{cs}}^2(t,k)-t^{-2}\right)dt+x^{-1}-\mathrm{cn}(x,k)\mathrm{ds}(x,k).$$

ⓘ

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

A&S Ref:

16.26.12

Referenced by:

§22.16(ii)

Permalink:

http://dlmf.nist.gov/22.16.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

…

39: 4.38 Inverse Hyperbolic Functions: Further Properties

… ►In the following equations square roots have their principal values. … ►All square roots have either possible value.

40: Bibliography E

… ►

On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

ⓘ

External Links:

Document, MathReview (S. Chowla), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

…