incomplete integrals

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21: 19.25 Relations to Other Functions

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19.25.13 $$D(\varphi ,k)=\frac{1}{3}{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\varphi$: 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 $$\mathrm{el2}(x,k_c,a,b)=a\mathrm{el1}(x,k_c)+\frac{1}{3}(b-a)R_D(r,r+k_c^2,r+1),$$

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Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind, $R_D(x,y,z)$: 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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19.25.22 $$\mathrm{el3}(x,k_c,p)=\mathrm{el1}(x,k_c)+\frac{1}{3}(1-p)R_J(r,r+k_c^2,r+1,r+p).$$

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Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

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19.25.25 $${(z-x)}^{3/2}{R}_D(x,y,z)=(3/k^2)(F(\varphi ,k)-E(\varphi ,k)),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(iii)

Permalink:

http://dlmf.nist.gov/19.25.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(iii), §19.25 and Ch.19

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19.25.27 $$2{(z-x)}^{-1/2}{R}_G(x,y,z)=E(\varphi ,k)+{(\cot \varphi )}^{2}F(\varphi ,k)+(\cot \varphi )\sqrt{1-k^2{\sin }^2\varphi }.$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\cot z$: cotangent function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(iii)

Permalink:

http://dlmf.nist.gov/19.25.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(iii), §19.25 and Ch.19

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22: 19.14 Reduction of General Elliptic Integrals

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19.14.1 $${\int }_1^x\frac{dt}{\sqrt{t^3-1}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}+1-x}{\sqrt{3}-1+x}}$, $k^2={\displaystyle \frac{2-\sqrt{3}}{4}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.2 $${\int }_x^1\frac{dt}{\sqrt{1-t^3}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}-1+x}{\sqrt{3}+1-x}}$, $k^2={\displaystyle \frac{2+\sqrt{3}}{4}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.3 $${\int }_0^x\frac{dt}{\sqrt{1+t^4}}=\frac{\mathrm{sign}\left(x\right)}{2}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{1-x^2}{1+x^2}}$, $k^2={\displaystyle \frac{1}{2}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\mathrm{sign}x$: sign of, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.4 $${\int }_y^x\frac{dt}{\sqrt{(a_1+b_1{t}^2)(a_2+b_2{t}^2)}}=\frac{1}{\sqrt{\gamma -\alpha }}F(\varphi ,k),$$ $k^2=(\gamma -\beta )/(\gamma -\alpha )$.

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Symbols:

$dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i), §19.14(i), §19.14(i), §19.15, §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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23: 19.8 Quadratic Transformations

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19.8.14 $$2(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{{\omega }^2-{\alpha }^2}{1+k^{\prime }}\mathrm{\Pi }({\varphi }_1,{\alpha }_1^2,k_1)+k^{2}F(\varphi ,k)-(1+k^{\prime }){\alpha }_1^2{R}_C(c_1,c_1-{\alpha }_1^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\varphi }_1$: change of variable, $\omega$: change of variable, ${\alpha }_1$: change of variable and $c_1$: change of variable

Permalink:

http://dlmf.nist.gov/19.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

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19.8.20 $$\rho \mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{4}{1+k^{\prime }}\mathrm{\Pi }({\psi }_1,{\alpha }_1^2,k_1)+(\rho -1)F(\varphi ,k)-R_C(c-1,c-{\alpha }^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\psi }_1$: change of variable, $\rho$: change of variable, ${\alpha }_1^2$: change of variable and $c$: change of variable

Referenced by:

§19.36(ii)

Permalink:

http://dlmf.nist.gov/19.8.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(iii), §19.8 and Ch.19

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24: 19.21 Connection Formulas

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§19.21(ii) Incomplete Integrals

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25: 19.36 Methods of Computation

… ►The incomplete integrals $R_F(x,y,z)$ and $R_G(x,y,z)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_C$, accompanied by two quadratically convergent series in the case of $R_G$; compare Carlson (1965, §§5,6). … ►Bulirsch (1969a, b) extend Bartky’s transformation to $\mathrm{el3}(x,k_c,p)$ by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. … ►For fast methods for computing the incomplete elliptic integral of the first kind see Karp and Sitnik (2007) and Fukushima (2010). …

26: 19.22 Quadratic Transformations

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§19.22(iii) Incomplete Integrals

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27: 19.33 Triaxial Ellipsoids

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19.33.2 $$\frac{S}{2\pi }=c^2+\frac{ab}{\sin \varphi }\left(E(\varphi ,k){\sin }^2\varphi +F(\varphi ,k){\cos }^2\varphi \right),$$ $a\ge b\ge c$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $S$: area

Permalink:

http://dlmf.nist.gov/19.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(i), §19.33 and Ch.19

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28: 8.8 Recurrence Relations and Derivatives

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8.8.14 $${\frac{\partial }{\partial a}{\gamma }^{\ast }(a,z)|}_{a=0}=-E_1\left(z\right)-\ln z.$$

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Symbols:

$E_1\left(z\right)$: exponential integral, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $a$: parameter

A&S Ref:

6.5.24

Permalink:

http://dlmf.nist.gov/8.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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29: 19.30 Lengths of Plane Curves

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19.30.3 $$s/a=E(\varphi ,k)=R_F(c-1,c-k^2,c)-\frac{1}{3}{k}^2{R}_D(c-1,c-k^2,c),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: 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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19.30.6 $$\frac{\partial s}{\partial (1/k)}=\sqrt{a^2-b^2}F(\varphi ,k)=\sqrt{a^2-b^2}{R}_F(c-1,c-k^2,c),$$ $k^2=(a^2-b^2)/(a^2+\lambda )$, $c={\csc }^2\varphi$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\csc z$: cosecant function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.30.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(i), §19.30 and Ch.19

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19.30.12 $$s=aF(\varphi ,1/\sqrt{2}),$$ $\varphi =\arcsin \sqrt{2/(q+1)}=\arccos \left(\tan \theta \right)$.

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\arccos z$: arccosine function, $\arcsin z$: arcsine function, $\tan z$: tangent function, $\varphi$: real or complex argument, $q$ and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(iii), §19.30 and Ch.19

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30: 19.5 Maclaurin and Related Expansions

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19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_3 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{F}_1(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};{\sin }^2\varphi ,{\alpha }^2{\sin }^2\varphi )k^{2m},$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.11))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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