19 Elliptic Integrals Symmetric Integrals 19.28 Integrals of Elliptic Integrals 19.30 Lengths of Plane Curves

§19.29 Reduction of General Elliptic Integrals

Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.29

§19.29(i) Reduction Theorems
§19.29(ii) Reduction to Basic Integrals
§19.29(iii) Examples

§19.29(i) Reduction Theorems

Notes:: For (19.29.4) see Carlson (1998, (3.6)). For (19.29.7), a special case of (19.29.8), see also Carlson (1987, (4.14)). For (19.29.8) see Carlson (1999, (4.10)) and Carlson (1988, (5.6)). For (19.29.10) see Byrd and Friedman (1971, p. 76, Eq. (234.13), and p. 74) for notation. Then use Carlson (2006b, (3.2)) with for reduction to $\mathop{R_{D}\/}\nolimits$ .
Keywords:: general elliptic integrals
Referenced by:: §19.15, §19.36(i)
Permalink:: http://dlmf.nist.gov/19.29.i

These theorems reduce integrals over a real interval of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over $(0,\infty)$ containing the square root of a cubic polynomial (compare §19.16(i)). Let

19.29.1

$X_{\alpha}=\sqrt{a_{\alpha}+b_{\alpha}x},$

$Y_{\alpha}=\sqrt{a_{\alpha}+b_{\alpha}y},$

, $1\leq\alpha\leq 5$ ,

Symbols:: $X_{\alpha}$ , $Y_{\alpha}$ and $\alpha$ : parameter
Referenced by:: §19.29(ii)
Permalink:: http://dlmf.nist.gov/19.29.E1
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19.29.2 $d_{{\alpha\beta}}=a_{\alpha}b_{\beta}-a_{\beta}b_{\alpha},$ $d_{{\alpha\beta}}\neq 0$ if $\alpha\neq\beta$ ,

Symbols:: $d_{{\alpha\beta}}$ , $\alpha$ : parameter and $\beta$ : parameter
Referenced by:: §19.29(ii)
Permalink:: http://dlmf.nist.gov/19.29.E2
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and assume that the line segment with endpoints $a_{\alpha}+b_{\alpha}x$ and $a_{\alpha}+b_{\alpha}y$ lies in $\Complex\setminus(-\infty,0)$ for $1\leq\alpha\leq 4$ . If

19.29.3 $s(t)=\prod_{{\alpha=1}}^{4}\sqrt{a_{\alpha}+b_{\alpha}t}$

Symbols:: : product and $\alpha$ : parameter
Referenced by:: §19.29(i)
Permalink:: http://dlmf.nist.gov/19.29.E3
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and $\alpha,\beta,\gamma,\delta$ is any permutation of the numbers , then

19.29.4 $\int_{y}^{x}\frac{dt}{s(t)}=2\!\mathop{R_{F}\/}\nolimits\!\left(U_{{12}}^{2},U% _{{13}}^{2},U_{{23}}^{2}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of , $\int$ : integral, $U_{n}$ and : product
Referenced by:: §19.28, §19.29(i), §19.29(i), §19.29(ii), §19.29(ii), §19.29(iii), §19.29(iii), §19.34, §19.36(i)
Permalink:: http://dlmf.nist.gov/19.29.E4
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where

19.29.5

$U_{{\alpha\beta}}=(X_{\alpha}X_{\beta}Y_{\gamma}Y_{\delta}+Y_{\alpha}Y_{\beta}% X_{\gamma}X_{\delta})/(x-y),$

$U_{{\alpha\beta}}=U_{{\beta\alpha}}=U_{{\gamma\delta}}=U_{{\delta\gamma}},$

$U_{{\alpha\beta}}^{2}-U_{{\alpha\gamma}}^{2}=d_{{\alpha\delta}}d_{{\beta\gamma% }}.$

Symbols:: $X_{\alpha}$ , $U_{n}$ , $Y_{\alpha}$ , $d_{{\alpha\beta}}$ , $\alpha$ : parameter, $\beta$ : parameter, $\gamma$ : parameter and $\delta$ : parameter
Permalink:: http://dlmf.nist.gov/19.29.E5
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There are only three distinct ’s with subscripts $\leq 4$ , and at most one of them can be 0 because the ’s are nonzero. Then

19.29.6

${U_{{\alpha\beta}}=\sqrt{b_{\alpha}}\sqrt{b_{\beta}}Y_{\gamma}Y_{\delta}+Y_{% \alpha}Y_{\beta}\sqrt{b_{\gamma}}\sqrt{b_{\delta}},}$ $x=\infty$ ,

$U_{{\alpha\beta}}=X_{\alpha}X_{\beta}\sqrt{-b_{\gamma}}\sqrt{-b_{\delta}}+% \sqrt{-b_{\alpha}}\sqrt{-b_{\beta}}X_{\gamma}X_{\delta},$ $y=-\infty$ .

Symbols:: $X_{\alpha}$ , $U_{n}$ , $Y_{\alpha}$ , $\alpha$ : parameter, $\beta$ : parameter, $\gamma$ : parameter and $\delta$ : parameter
Permalink:: http://dlmf.nist.gov/19.29.E6
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19.29.7 $\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{\delta}+b_{\delta}t}\frac{dt}{s(t% )}=\tfrac{2}{3}d_{{\alpha\beta}}d_{{\alpha\gamma}}\mathop{R_{D}\/}\nolimits\!% \left(U_{{\alpha\beta}}^{2},U_{{\alpha\gamma}}^{2},U_{{\alpha\delta}}^{2}% \right)+\frac{2X_{\alpha}Y_{\alpha}}{X_{{\delta}}Y_{\delta}U_{{\alpha\delta}}},$ $U_{{\alpha\delta}}\neq 0$ .

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, : differential of , $\int$ : integral, $X_{\alpha}$ , $U_{n}$ , $Y_{\alpha}$ , $d_{{\alpha\beta}}$ , : 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
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19.29.8 $\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{5}+b_{5}t}\frac{dt}{s(t)}=\frac{2% }{3}\frac{d_{{\alpha\beta}}d_{{\alpha\gamma}}d_{{\alpha\delta}}}{d_{{\alpha 5}% }}\mathop{R_{J}\/}\nolimits\!\left(U_{{12}}^{2},U_{{13}}^{2},U_{{23}}^{2},U_{{% \alpha 5}}^{2}\right)+2\!\mathop{R_{C}\/}\nolimits\!\left(S_{{\alpha 5}}^{2},Q% _{{\alpha 5}}^{2}\right),$ $S_{{\alpha 5}}^{2}\in\Complex\setminus(-\infty,0)$ ,

where

19.29.9

$U_{{\alpha 5}}^{2}=U_{{\alpha\beta}}^{2}-\frac{d_{{\alpha\gamma}}d_{{\alpha% \delta}}d_{{\beta 5}}}{d_{{\alpha 5}}}=U_{{\beta\gamma}}^{2}-\frac{d_{{\alpha% \beta}}d_{{\alpha\gamma}}d_{{\delta 5}}}{d_{{\alpha 5}}}\neq 0,$

$S_{{\alpha 5}}=\frac{1}{x-y}\left(\frac{X_{\beta}X_{\gamma}X_{\delta}}{X_{% \alpha}}Y_{5}^{2}+\frac{Y_{\beta}Y_{\gamma}Y_{\delta}}{Y_{\alpha}}X_{5}^{2}% \right),$

$Q_{{\alpha 5}}=\frac{X_{5}Y_{5}}{X_{\alpha}Y_{\alpha}}U_{{\alpha 5}}\neq 0,$

$S_{{\alpha 5}}^{2}-Q_{{\alpha 5}}^{2}=\frac{d_{{\beta 5}}d_{{\gamma 5}}d_{{% \delta 5}}}{d_{{\alpha 5}}}.$

Symbols:: $X_{\alpha}$ , $U_{n}$ , $S_{n}$ , $Q_{{\alpha\beta}}$ , $Y_{\alpha}$ , $d_{{\alpha\beta}}$ , $\alpha$ : parameter, $\beta$ : parameter, $\gamma$ : parameter and $\delta$ : parameter
Permalink:: http://dlmf.nist.gov/19.29.E9
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The Cauchy principal value is taken when $U_{{\alpha 5}}^{2}$ or $Q_{{\alpha 5}}^{2}$ is real and negative. Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1.

The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking $x^{2}$ as the variable of integration in 3.152. Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. (19.29.7) subsumes all 72 formulas in Gradshteyn and Ryzhik (2000, 3.168), and its cubic cases similarly replace the formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.142, and 3.141(1-18)). For example, 3.142(2) is included as

19.29.10 $\int_{u}^{b}\sqrt{\frac{a-t}{(b-t)(t-c)^{3}}}dt=-\tfrac{2}{3}{(a-b)}{(b-u)}^{{% 3/2}}\mathop{R_{D}\/}\nolimits+\frac{2}{b-c}\sqrt{\frac{(a-u)(b-u)}{u-c}},$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, : differential of and $\int$ : integral
Referenced by:: §19.29(i)
Permalink:: http://dlmf.nist.gov/19.29.E10
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where the arguments of the $\mathop{R_{D}\/}\nolimits$ function are, in order, , , .

§19.29(ii) Reduction to Basic Integrals

Keywords:: basic elliptic integrals, general elliptic integrals
Referenced by:: §19.15, §19.34
Permalink:: http://dlmf.nist.gov/19.29.ii

(19.2.3) can be written

19.29.11 $I(\mathbf{m})=\int_{y}^{x}\prod_{{\alpha=1}}^{h}(a_{\alpha}+b_{\alpha}t)^{{-1/% 2}}\prod_{{j=1}}^{n}(a_{j}+b_{j}t)^{{m_{j}}}dt,$

Symbols:: : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, $I(\mathbf{m})$ : integral and $\alpha$ : parameter
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.29.E11
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where , or 4, $n\geq h$ , and $m_{j}$ is an integer. Define

19.29.12 $\mathbf{m}=(m_{1},\dots,m_{n})=\sum_{{j=1}}^{n}m_{j}\mathbf{e}_{j},$

Symbols:: : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/19.29.E12
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where $\mathbf{e}_{j}$ is an -tuple with 1 in the th position and 0’s elsewhere. Define also $\boldsymbol{{0}}=(0,\dots,0)$ and retain the notation and conditions associated with (19.29.1) and (19.29.2). The integrals in (19.29.4), (19.29.7), and (19.29.8) are $I(\boldsymbol{{0}})$ , $I(\mathbf{e}_{\alpha}-\mathbf{e}_{\delta})$ , and $I(\mathbf{e}_{\alpha}-\mathbf{e}_{5})$ , respectively.

The only cases of $I(\mathbf{m})$ that are integrals of the first kind are the two ( or 4) with $\mathbf{m}=\boldsymbol{{0}}$ . The only cases that are integrals of the third kind are those in which at least one $m_{j}$ with is a negative integer and those in which and $\sum_{{j=1}}^{n}m_{j}$ is a positive integer. All other cases are integrals of the second kind.

$I(\mathbf{m})$ can be reduced to a linear combination of basic integrals and algebraic functions. In the cubic case () the basic integrals are

19.29.13

$I(\boldsymbol{{0}});$

$I(-\mathbf{e}_{j}),$ $1\leq j\leq n$ .

Symbols:: : nonnegative integer and $I(\mathbf{m})$ : integral
Permalink:: http://dlmf.nist.gov/19.29.E13
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In the quartic case () the basic integrals are

19.29.14

$I(\boldsymbol{{0}});$

$I(-\mathbf{e}_{j}),$ $1\leq j\leq n$ ;

$I(\mathbf{e}_{\alpha}),$ $1\leq\alpha\leq 4$ .

Symbols:: : nonnegative integer, $I(\mathbf{m})$ : integral and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/19.29.E14
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

Basic integrals of type $I(-\mathbf{e}_{j})$ , $1\leq j\leq h$ , are not linearly independent, nor are those of type $I(\mathbf{e}_{j})$ , $1\leq j\leq 4$ .

The reduction of $I(\mathbf{m})$ is carried out by a relation derived from partial fractions and by use of two recurrence relations. These are given in Carlson (1999, (2.19), (3.5), (3.11)) and simplified in Carlson (2002, (1.10), (1.7), (1.8)) by means of modified definitions. Partial fractions provide a reduction to integrals in which $\mathbf{m}$ has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. A special case of Carlson (1999, (2.19)) is given by

19.29.15 $b_{j}I(\mathbf{e}_{l}-\mathbf{e}_{j})=d_{{lj}}I(-\mathbf{e}_{j})+b_{l}I(% \boldsymbol{{0}}),$ $j,l=1,2,\dots,n$ ,

Symbols:: : nonnegative integer, : nonnegative integer, $I(\mathbf{m})$ : integral and $d_{{\alpha\beta}}$
Referenced by:: §19.15, §19.29(ii), §19.34
Permalink:: http://dlmf.nist.gov/19.29.E15
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which shows how to express the basic integral $I(-\mathbf{e}_{j})$ in terms of symmetric integrals by using (19.29.4) and either (19.29.7) or (19.29.8). The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting $x^{2}=t$ in some cases).

If , then the recurrence relation (Carlson (1999, (3.5))) has the special case

19.29.16 $b_{\beta}b_{\gamma}I(\mathbf{e}_{\alpha})=d_{{\alpha\beta}}d_{{\alpha\gamma}}I% (-\mathbf{e}_{\alpha})+2b_{\alpha}\left(\frac{s(x)}{a_{\alpha}+b_{\alpha}x}-% \frac{s(y)}{a_{\alpha}+b_{\alpha}y}\right),$

Symbols:: $I(\mathbf{m})$ : integral, : product, $d_{{\alpha\beta}}$ , $\alpha$ : parameter, $\beta$ : parameter and $\gamma$ : parameter
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.29.E16
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where $\alpha,\beta,\gamma$ is any permutation of the numbers , and

19.29.17 $s(t)=\prod_{{\alpha=1}}^{3}\sqrt{a_{\alpha}+b_{\alpha}t}.$

Symbols:: : product and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/19.29.E17
Encodings:: TeX, pMML, png

(This shows why $I(\mathbf{e}_{\alpha})$ is not needed as a basic integral in the cubic case.) In the quartic case this recurrence relation has an extra term in $I(2\mathbf{e}_{\alpha})$ , and hence $I(\mathbf{e}_{\alpha})$ , $1\leq\alpha\leq 4$ , is a basic integral. It can be expressed in terms of symmetric integrals by setting $a_{5}=1$ and $b_{5}=0$ in (19.29.8).

The other recurrence relation is

19.29.18 $b_{j}^{q}I(q\mathbf{e}_{l})=\sum_{{r=0}}^{q}\binom{q}{r}b_{l}^{r}d_{{lj}}^{{q-% r}}I(r\mathbf{e}_{j}),$ $j,l=1,2,\dots,n$ ;

Symbols:: $\binom{m}{n}$ : binomial coefficient, : nonnegative integer, : nonnegative integer, $I(\mathbf{m})$ : integral and $d_{{\alpha\beta}}$
Referenced by:: §19.29(ii), §19.34
Permalink:: http://dlmf.nist.gov/19.29.E18
Encodings:: TeX, pMML, png

see Carlson (1999, (3.11)). An example that uses (19.29.15)–(19.29.18) is given in §19.34.

For an implementation by James FitzSimons of the method for reducing $I(\mathbf{m})$ to basic integrals and extensive tables of such reductions, see Carlson (1999) and Carlson and FitzSimons (2000).

Another method of reduction is given in Gray (2002). It depends primarily on multivariate recurrence relations that replace one integral by two or more.

§19.29(iii) Examples

Notes:: Take $t^{2}$ as a new variable where appropriate. Then factor quadratic polynomials, use (19.29.4), and apply (19.22.18) to remove any complex quantities. For (19.29.20) use (19.29.7) with $a_{{\alpha}}+b_{{\alpha}}t=t$ and $a_{{\delta}}+b_{{\delta}}t=1$ . For (19.29.21) use (19.29.7) with $a_{{\alpha}}+b_{{\alpha}}t=1$ and $a_{{\delta}}+b_{{\delta}}t=t$ . With regard to (19.29.28) see Carlson (1977a, p. 238).
Keywords:: general elliptic integrals, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.29.iii

The first formula replaces (19.14.4)–(19.14.10). Define $Q_{j}(t)=a_{j}+b_{j}t^{2}$ , and assume both ’s are positive for $0\leq y<t<x$ . Then

19.29.19 $\int_{y}^{x}\frac{dt}{\sqrt{Q_{1}(t)Q_{2}(t)}}=\mathop{R_{F}\/}\nolimits\!% \left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of , $\int$ : integral, and $Q(t^{2})$
Referenced by:: §19.14(i), §19.15, §19.20(i), §19.25(v)
Permalink:: http://dlmf.nist.gov/19.29.E19
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19.29.20 $\int_{y}^{x}\frac{t^{2}dt}{\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}a_{1}a_{2}% \mathop{R_{D}\/}\nolimits\!\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right% )+(xy/U),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, : differential of , $\int$ : integral, 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
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and

19.29.21 $\int_{y}^{x}\frac{dt}{t^{2}\sqrt{Q_{1}(t)Q_{2}(t)}}=\tfrac{1}{3}b_{1}b_{2}% \mathop{R_{D}\/}\nolimits\!\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right% )+(xyU)^{{-1}},$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, : differential of , $\int$ : integral, and $Q(t^{2})$
Referenced by:: §19.25(v), §19.29(iii), §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E21
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where

19.29.22 $(x^{2}-y^{2})U=x\sqrt{Q_{1}(y)Q_{2}(y)}+y\sqrt{Q_{1}(x)Q_{2}(x)}.$

Symbols:: and $Q(t^{2})$
Referenced by:: §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E22
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If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $\mathop{R_{G}\/}\nolimits\!\left(a_{1}b_{2},a_{2}b_{1},0\right)$ multiplied either by $-2/(b_{1}b_{2})$ or by $-2/(a_{1}a_{2})$ in the cases of (19.29.20) or (19.29.21), respectively. If $x=\infty$ , then is found by taking the limit. For example,

19.29.23 $\int_{y}^{{\infty}}\frac{dt}{\sqrt{(t^{2}+a^{2})(t^{2}-b^{2})}}=\mathop{R_{F}% \/}\nolimits\!\left(y^{2}+a^{2},y^{2}-b^{2},y^{2}\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/19.29.E23
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Next, for , define $Q_{j}(t)=f_{j}+g_{j}t+h_{j}t^{2}$ , and assume both ’s are positive for . If each has real zeros, then (19.29.4) may be simpler than

19.29.24 $\int_{y}^{x}\frac{dt}{\sqrt{Q_{1}(t)Q_{2}(t)}}=4\!\mathop{R_{F}\/}\nolimits\!% \left(U,U+D_{{12}}+V,U+D_{{12}}-V\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of , $\int$ : integral, , $D_{{jl}}$ , and $Q(t^{2})$
Permalink:: http://dlmf.nist.gov/19.29.E24
Encodings:: TeX, pMML, png

where

19.29.25

$(x-y)^{2}U=S_{1}S_{2},$

$S_{j}=\left(\sqrt{Q_{j}(x)}+\sqrt{Q_{j}(y)}\right)^{2}-h_{j}(x-y)^{2},$

$D_{{jl}}=2f_{j}h_{l}+2h_{j}f_{l}-g_{j}g_{l},$

$V=\sqrt{D_{{12}}^{2}-D_{{11}}D_{{22}}}.$

Symbols:: : nonnegative integer, , $D_{{jl}}$ , $S_{j}$ , , $Q(t^{2})$ , : coefficient, : coefficient and : coefficient
Referenced by:: §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.29.E25
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(The variables of $\mathop{R_{F}\/}\nolimits$ are real and nonnegative unless both ’s have real zeros and those of $Q_{1}$ interlace those of $Q_{2}$ .) If $Q_{1}(t)=(a_{1}+b_{1}t)(a_{2}+b_{2}t)$ , where both linear factors are positive for , and $Q_{2}(t)=f_{2}+g_{2}t+h_{2}t^{2}$ , then (19.29.25) is modified so that