two variables

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11—20 of 161 matching pages

11: 19.25 Relations to Other Functions

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19.25.6

\frac{\partial F\left(\phi,k\right)}{\partial k}=\tfrac{1}{3}kR_{D}\left(c-1,c% ,c-k^{2}\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i)
Permalink:: http://dlmf.nist.gov/19.25.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(i), §19.25 and Ch.19

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19.25.9

E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}% \left(c-1,c-k^{2},c\right),

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i), §19.25(i), §19.25(iii), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.25.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(i), §19.25 and Ch.19

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19.25.12

\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,% c-k^{2},c\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i)
Permalink:: http://dlmf.nist.gov/19.25.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(i), §19.25 and Ch.19

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19.25.13

D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $D\left(\NVar{\phi},\NVar{k}\right)$ : incomplete elliptic integral of Legendre’s type, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.25.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(i), §19.25 and Ch.19

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19.25.21

\operatorname{el2}\left(x,k_{c},a,b\right)=a\operatorname{el1}\left(x,k_{c}% \right)+\tfrac{1}{3}{(b-a)}R_{D}\left(r,r+k_{c}^{2},r+1\right),

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Symbols:: $\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\operatorname{el2}\left(\NVar{x},\NVar{k_{c}},\NVar{a},\NVar{b}\right)$ : Bulirsch’s incomplete elliptic integral of the second kind, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $k$ : real or complex modulus and $r$ : inverse
Permalink:: http://dlmf.nist.gov/19.25.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(ii), §19.25 and Ch.19

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12: 19.34 Mutual Inductance of Coaxial Circles

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19.34.5

\frac{3{c}^{2}}{8\pi ab}M=3R_{F}\left(0,r_{+}^{2},r_{-}^{2}\right)-2r_{-}^{2}R% _{D}\left(0,r_{+}^{2},r_{-}^{2}\right),

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $M$ : mutual inductance and $r_{\pm}$
Permalink:: http://dlmf.nist.gov/19.34.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.34 and Ch.19

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13: 16.16 Transformations of Variables

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14: 19.17 Graphics

… ►Because the

R

-function is homogeneous, there is no loss of generality in giving one variable the value

1

-1

(as in Figure 19.3.2). For

R_{F}

R_{G}

, and

R_{J}

, which are symmetric in

x, y, z

, we may further assume that

z

is the largest of

x, y, z

if the variables are real, then choose

z=1

, and consider only

0\leq x\leq 1

and

0\leq y\leq 1

. …

15: 19.30 Lengths of Plane Curves

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19.30.3

s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

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19.30.9

s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},

r=b^{4}/y^{2}

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $I(\mathbf{m})$ : integral and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

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16: 19.22 Quadratic Transformations

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19.22.3

2y^{2}R_{D}\left(0,x^{2},y^{2}\right)=\tfrac{1}{4}(y^{2}-x^{2})R_{D}\left(0,xy% ,a^{2}\right)+3R_{F}\left(0,xy,a^{2}\right).

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(i), §19.22(i), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(i), §19.22 and Ch.19

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19.22.10

R_{D}\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_{0}\right)% a_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.22(ii)
Permalink:: http://dlmf.nist.gov/19.22.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(ii), §19.22 and Ch.19

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19.22.19

(z_{\pm}^{2}-z_{\mp}^{2})R_{D}\left(x^{2},y^{2},z^{2}\right)={2(z_{\pm}^{2}-a^% {2})}R_{D}\left(a^{2},z_{\mp}^{2},z_{\pm}^{2}\right)-3R_{F}\left(x^{2},y^{2},z% ^{2}\right)+(3/z),

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.22(iii), §19.22(iii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §19.22(iii), §19.22 and Ch.19

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17: 19.29 Reduction of General Elliptic Integrals

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19.29.7

\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{\delta}+b_{\delta}t}\frac{\,% \mathrm{d}t}{s(t)}=\tfrac{2}{3}d_{\alpha\beta}d_{\alpha\gamma}R_{D}\left(U_{% \alpha\beta}^{2},U_{\alpha\gamma}^{2},U_{\alpha\delta}^{2}\right)+\frac{2X_{% \alpha}Y_{\alpha}}{X_{\delta}Y_{\delta}U_{\alpha\delta}},

U_{\alpha\delta}\neq 0

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $X_{\alpha}$ , $U_{n}$ , $Y_{\alpha}$ , $d_{\alpha\beta}$ , $s(t)$ : product, $\alpha$ : parameter, $\beta$ : parameter, $\gamma$ : parameter and $\delta$ : parameter
Referenced by:: §19.29(i), §19.29(i), §19.29(ii), §19.29(ii), §19.29(iii), §19.30(ii), §19.34
Permalink:: http://dlmf.nist.gov/19.29.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(i), §19.29 and Ch.19

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19.29.10

\int_{u}^{b}\sqrt{\frac{a-t}{(b-t)(t-c)^{3}}}\,\mathrm{d}t=-\tfrac{2}{3}{(a-b)% }{(b-u)}^{3/2}R_{D}+\frac{2}{b-c}\sqrt{\frac{(a-u)(b-u)}{u-c}},

a>b>u>c

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.29(i)
Permalink:: http://dlmf.nist.gov/19.29.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(i), §19.29 and Ch.19

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19.29.20

\int_{y}^{x}\frac{t^{2}\,\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}a_{% 1}a_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)+(xy/U),

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $U$ and $Q(t^{2})$
Referenced by:: §19.20(iv), §19.25(v), §19.29(iii), §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

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19.29.21

\int_{y}^{x}\frac{\,\mathrm{d}t}{t^{2}\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}b_{% 1}b_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)+(xyU)^{-1},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $U$ and $Q(t^{2})$
Referenced by:: §19.25(v), §19.29(iii), §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.29(iii), §19.29 and Ch.19

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18: 19.1 Special Notation

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19: 19.33 Triaxial Ellipsoids

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19.33.7

L_{c}=2\pi abc\int_{0}^{\infty}\frac{\,\mathrm{d}\lambda}{\sqrt{(a^{2}+\lambda% )(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left(a^{2},b^{2},c^{2}\right).

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L_{a}$ , $L_{b}$ : factor and $V$ : volume
Referenced by:: §19.15, §19.33(iii)
Permalink:: http://dlmf.nist.gov/19.33.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(iii), §19.33 and Ch.19

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20: 19.26 Addition Theorems

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19.26.7

R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+R_{D}\left(x+\mu,y+\mu,z+\mu% \right)=R_{D}\left(x,y,z\right)-\frac{3}{\sqrt{z(z+\lambda)(z+\mu)}},

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, $x$ : positive, $y$ : positive, $z$ : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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19.26.20

R_{D}\left(x,y,z\right)=2R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+\frac% {3}{\sqrt{z}(z+\lambda)}.

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

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