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Specific values of Carlson symmetric elliptic integrals

Table of contents: The elementary integral RC - The integral of the first kind RF - The integral of the second kind RG - The degenerate integral of the third kind RD - The integral of the third kind RJ

Related topics: Carlson symmetric elliptic integrals

The elementary integral RC

Scale invariance

a839d5
RC ⁣(λx,λy)=λ1/2RC ⁣(x,y)R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right)

Particular constant values

5c2b08
RC ⁣(0,0)=~R_C\!\left(0, 0\right) = {\tilde \infty}
1acb07
RC ⁣(1,0)=R_C\!\left(1, 0\right) = \infty
e464ec
RC ⁣(0,1)=π2R_C\!\left(0, 1\right) = \frac{\pi}{2}
d38c27
RC ⁣(1,1)=1R_C\!\left(1, 1\right) = 1
eac389
RC ⁣(1,2)=π4R_C\!\left(1, 2\right) = \frac{\pi}{4}
a15c03
RC ⁣(2,1)=log ⁣(1+2)R_C\!\left(2, 1\right) = \log\!\left(1 + \sqrt{2}\right)
35cb93
RC ⁣(0,1)=πi2R_C\!\left(0, -1\right) = -\frac{\pi i}{2}
56d1bc
RC ⁣(1,0)=iR_C\!\left(-1, 0\right) = -i \infty
25435b
RC ⁣(1,1)=2log ⁣(1+2)2π24iR_C\!\left(1, -1\right) = \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} - \frac{\pi \sqrt{2}}{4} i
7ea1ad
RC ⁣(1,1)=π242log ⁣(1+2)2iR_C\!\left(-1, 1\right) = \frac{\pi \sqrt{2}}{4} - \frac{\sqrt{2} \log\!\left(1 + \sqrt{2}\right)}{2} i

Specialized values

7cbe17
RC ⁣(x,0)={sgn ⁣(1x),x0~,x=0R_C\!\left(x, 0\right) = \begin{cases} \operatorname{sgn}\!\left(\frac{1}{\sqrt{x}}\right) \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}
ff58cf
RC ⁣(0,y)={π2y,y0~,y=0R_C\!\left(0, y\right) = \begin{cases} \frac{\pi}{2 \sqrt{y}}, & y \ne 0\\{\tilde \infty}, & y = 0\\ \end{cases}
ad96f4
RC ⁣(x,x)=1xR_C\!\left(x, x\right) = \frac{1}{\sqrt{x}}
09a494
RC ⁣(x,2x)=π4xR_C\!\left(x, 2 x\right) = \frac{\pi}{4 \sqrt{x}}
b136bd
RC ⁣(2x,x)=log ⁣(1+2)xR_C\!\left(2 x, x\right) = \frac{\log\!\left(1 + \sqrt{2}\right)}{\sqrt{x}}

General formulas for real variables

5ada5f
RC ⁣(x,y)={atan ⁣(yx1)yx,x<y1x,x=yatanh ⁣(1yx)xy,x>yR_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}
718f3a
RC ⁣(x,y)={acos ⁣(xy)yx,x<y1x,x=yacosh ⁣(xy)xy,x>yR_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{acos}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{y - x}}, & x < y\\\frac{1}{\sqrt{x}}, & x = y\\\frac{\operatorname{acosh}\!\left(\sqrt{\frac{x}{y}}\right)}{\sqrt{x - y}}, & x > y\\ \end{cases}
de0638
RC ⁣(x,y)=iRC ⁣(x,y)R_C\!\left(-x, -y\right) = -i R_C\!\left(x, y\right)
00cdb7
RC ⁣(x,y)=1x+y(atanh ⁣(xx+y)πi2)R_C\!\left(x, -y\right) = \frac{1}{\sqrt{x + y}} \left(\operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) - \frac{\pi i}{2}\right)
bc2f88
RC ⁣(x,y)=1x+y(π2atanh ⁣(xx+y)i)R_C\!\left(-x, y\right) = \frac{1}{\sqrt{x + y}} \left(\frac{\pi}{2} - \operatorname{atanh}\!\left(\sqrt{\frac{x}{x + y}}\right) i\right)
4becdd
RC ⁣(x,y)=iRC ⁣(x,y)R_C\!\left(-x, y\right) = \overline{i R_C\!\left(x, -y\right)}

General formulas for one or more complex variables

7b5755
RC ⁣(x,y)={atan ⁣(yx1)yx,xy1x,x=yR_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{y}{x} - 1}\right)}{\sqrt{y - x}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}
0cf60d
RC ⁣(x,y)={atanh ⁣(1yx)xy,xy1x,x=yR_C\!\left(x, y\right) = \begin{cases} \frac{\operatorname{atanh}\!\left(\sqrt{1 - \frac{y}{x}}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}
8c9ba1
RC ⁣(x,cx)={atan ⁣(c1)(c1)x,c>11x,c=1atanh ⁣(1c)(1c)x,c<1R_C\!\left(x, c x\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{c - 1}\right)}{\sqrt{\left(c - 1\right) x}}, & c > 1\\\frac{1}{\sqrt{x}}, & c = 1\\\frac{\operatorname{atanh}\!\left(\sqrt{1 - c}\right)}{\sqrt{\left(1 - c\right) x}}, & c < 1\\ \end{cases}
7348e3
RC ⁣(x,cx)=1(c+1)x{atanh ⁣(c+1),Im(x)<0  or  (Im(x)=0  and  Re(x)0)atanh ⁣(c+1)+πi,otherwiseR_C\!\left(x, -c x\right) = \frac{1}{\sqrt{\left(c + 1\right) x}} \begin{cases} \operatorname{atanh}\!\left(\sqrt{c + 1}\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\\operatorname{atanh}\!\left(\sqrt{c + 1}\right) + \pi i, & \text{otherwise}\\ \end{cases}
eb1d4f
RC ⁣(1,1+y)={atan ⁣(y)y,y01,y=0R_C\!\left(1, 1 + y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{y}\right)}{\sqrt{y}}, & y \ne 0\\1, & y = 0\\ \end{cases}
157ebb
RC ⁣(1,1+y)=2F1 ⁣(1,12,32,y)R_C\!\left(1, 1 + y\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, -y\right)

The integral of the first kind RF

Symmetry and scale invariance

f29729
RF ⁣(x,y,z)=RF ⁣(x,z,y)=RF ⁣(y,x,z)=RF ⁣(y,z,x)=RF ⁣(z,x,y)=RF ⁣(z,y,x)R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right)
7a168a
RF ⁣(λx,λy,λz)=λ1/2RF ⁣(x,y,z)R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right)

Particular constant values

e39456
RF ⁣(0,0,0)=~R_F\!\left(0, 0, 0\right) = {\tilde \infty}
9a95a5
RF ⁣(0,0,1)=R_F\!\left(0, 0, 1\right) = \infty
8bb972
RF ⁣(0,1,1)=π2R_F\!\left(0, 1, 1\right) = \frac{\pi}{2}
c166ca
RF ⁣(1,1,1)=1R_F\!\left(1, 1, 1\right) = 1
4cd504
RF ⁣(1,1,2)=log ⁣(1+2)R_F\!\left(1, 1, 2\right) = \log\!\left(1 + \sqrt{2}\right)
0bf328
RF ⁣(1,2,2)=π4R_F\!\left(1, 2, 2\right) = \frac{\pi}{4}
28237a
RF ⁣(0,1,2)=(Γ ⁣(14))242πR_F\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}
f1dd8a
RF ⁣(0,1,1)=(Γ ⁣(14))242π(1i)R_F\!\left(0, 1, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right)
4c1988
RF ⁣(0,2,4)=(Γ ⁣(14))28πR_F\!\left(0, 2, 4\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}}
6c4567
RF ⁣(0,12,1)=(Γ ⁣(14))24πR_F\!\left(0, \frac{1}{2}, 1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}
90af98
RF ⁣(0,0,1)=iR_F\!\left(0, 0, -1\right) = -i \infty
3a84d6
RF ⁣(0,1,1)=πi2R_F\!\left(0, -1, -1\right) = \frac{-\pi i}{2}
6674bb
RF ⁣(1,1,1)=iR_F\!\left(-1, -1, -1\right) = -i
5c178f
RF ⁣(0,1,2)=(Γ ⁣(14))242πiR_F\!\left(0, -1, -2\right) = -\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} i
e30d7e
RF ⁣(0,1,12216)=(2+2)(Γ ⁣(14))216πR_F\!\left(0, 1, 12 \sqrt{2} - 16\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}}
cf5caa
RF ⁣(0,i,i)=(Γ ⁣(14))24πR_F\!\left(0, i, -i\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}}
67e015
RF ⁣(0,(Γ ⁣(14))416π,(Γ ⁣(14))432π)=1R_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{16 \pi}, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1
8519dd
RF ⁣(0,(Γ ⁣(14))432π,(Γ ⁣(14))432π)=1iR_F\!\left(0, \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}, \frac{-{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{32 \pi}\right) = 1 - i

Special parametric cases

13a092
RF ⁣(0,0,x)={,x0~,x=0R_F\!\left(0, 0, x\right) = \begin{cases} \infty, & x \ne 0\\{\tilde \infty}, & x = 0\\ \end{cases}
53d869
RF ⁣(0,1,x)=K ⁣(1x)R_F\!\left(0, 1, x\right) = K\!\left(1 - x\right)
ab5af3
RF ⁣(0,x,x)=π2xR_F\!\left(0, x, x\right) = \frac{\pi}{2 \sqrt{x}}
415ff0
RF ⁣(0,x,y)=K ⁣(1yx)xR_F\!\left(0, x, y\right) = \frac{K\!\left(1 - \frac{y}{x}\right)}{\sqrt{x}}
0ed5e2
RF ⁣(0,x,2x)=1x(Γ ⁣(14))242πR_F\!\left(0, x, 2 x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}}
e54e61
RF ⁣(0,x,x)=1x(Γ ⁣(14))242π{1i,Im(x)<0  or  (Im(x)=0  and  Re(x)0)1+i,otherwiseR_F\!\left(0, x, -x\right) = \frac{1}{\sqrt{x}} \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \begin{cases} 1 - i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 + i, & \text{otherwise}\\ \end{cases}
538c8c
RF ⁣(0,x,cx)=K ⁣(1c)xR_F\!\left(0, x, c x\right) = \frac{K\!\left(1 - c\right)}{\sqrt{x}}
271b73
RF ⁣(0,x,cx)=1x{K ⁣(1+c),Im(x)<0  or  (Im(x)=0  and  Re(x)0)K ⁣(1+c)+2iK ⁣(c),otherwiseR_F\!\left(0, x, -c x\right) = \frac{1}{\sqrt{x}} \begin{cases} K\!\left(1 + c\right), & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\K\!\left(1 + c\right) + 2 i K\!\left(-c\right), & \text{otherwise}\\ \end{cases}
63d11e
RF ⁣(x,y,y)=RC ⁣(x,y)R_F\!\left(x, y, y\right) = R_C\!\left(x, y\right)
ebaa1a
RF ⁣(x,x,y)=RC ⁣(y,x)R_F\!\left(x, x, y\right) = R_C\!\left(y, x\right)
649dc0
RF ⁣(x,x,y)={atan ⁣(xy1)xy,xy1x,x=yR_F\!\left(x, x, y\right) = \begin{cases} \frac{\operatorname{atan}\!\left(\sqrt{\frac{x}{y} - 1}\right)}{\sqrt{x - y}}, & x \ne y\\\frac{1}{\sqrt{x}}, & x = y\\ \end{cases}
9b0388
RF ⁣(x,x,x)=1xR_F\!\left(x, x, x\right) = \frac{1}{\sqrt{x}}
5ab6bf
RF ⁣(x,y,z)=iRF ⁣(x,y,z)R_F\!\left(-x, -y, -z\right) = -i R_F\!\left(x, y, z\right)
23e0a7
RF ⁣(x,y,z)=iRF ⁣(x,y,z)R_F\!\left(-x, -y, z\right) = \overline{i R_F\!\left(x, y, -z\right)}

The integral of the second kind RG

Symmetry and scale invariance

b478a1
RG ⁣(x,y,z)=RG ⁣(x,z,y)=RG ⁣(y,x,z)=RG ⁣(y,z,x)=RG ⁣(z,x,y)=RG ⁣(z,y,x)R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right)
f9ca94
RG ⁣(λx,λy,λz)=λ1/2RG ⁣(x,y,z)R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right)

Particular constant values

bcc121
RG ⁣(0,0,0)=0R_G\!\left(0, 0, 0\right) = 0
d5ff09
RG ⁣(0,0,1)=12R_G\!\left(0, 0, 1\right) = \frac{1}{2}
cd55cf
RG ⁣(0,1,1)=π4R_G\!\left(0, 1, 1\right) = \frac{\pi}{4}
250ff1
RG ⁣(1,1,1)=1R_G\!\left(1, 1, 1\right) = 1
4d7098
RG ⁣(1,1,2)=22+log ⁣(1+2)2R_G\!\left(1, 1, 2\right) = \frac{\sqrt{2}}{2} + \frac{\log\!\left(1 + \sqrt{2}\right)}{2}
d51efc
RG ⁣(1,2,2)=π4+12R_G\!\left(1, 2, 2\right) = \frac{\pi}{4} + \frac{1}{2}
84f403
RG ⁣(0,1,2)=(Γ ⁣(14))282π+π3/22(Γ ⁣(14))2R_G\!\left(0, 1, 2\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}
9e30e7
RG ⁣(0,1,1)=2π3/22(Γ ⁣(14))2(1+i)R_G\!\left(0, 1, -1\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right)
c5a9cf
RG ⁣(0,16,16)=πR_G\!\left(0, 16, 16\right) = \pi

Special parametric cases

d829be
RG ⁣(0,0,x)=x2R_G\!\left(0, 0, x\right) = \frac{\sqrt{x}}{2}
3f6d40
RG ⁣(0,1,x)=E ⁣(1x)2R_G\!\left(0, 1, x\right) = \frac{E\!\left(1 - x\right)}{2}
cdb587
RG ⁣(0,x,x)=πx4R_G\!\left(0, x, x\right) = \frac{\pi \sqrt{x}}{4}
7cddc6
RG ⁣(0,x,y)=xE ⁣(1yx)2R_G\!\left(0, x, y\right) = \frac{\sqrt{x} E\!\left(1 - \frac{y}{x}\right)}{2}
3f1547
RG ⁣(0,x,2x)=x((Γ ⁣(14))282π+π3/22(Γ ⁣(14))2)R_G\!\left(0, x, 2 x\right) = \sqrt{x} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{2 \pi}} + \frac{{\pi}^{3 / 2}}{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right)
7c50d1
RG ⁣(0,x,x)=x2π3/22(Γ ⁣(14))2{1+i,Im(x)<0  or  (Im(x)=0  and  Re(x)0)1i,otherwiseR_G\!\left(0, x, -x\right) = \sqrt{x} \frac{\sqrt{2} {\pi}^{3 / 2}}{2 {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \begin{cases} 1 + i, & \operatorname{Im}(x) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(x) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0\right)\\1 - i, & \text{otherwise}\\ \end{cases}