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From Other approaches to $\int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \text{d}x$, I want to know the transformation about $$ \int_0^1\frac{K(x)}{\sqrt{3-x}}dx=\frac{1}{\sqrt2}\int_0^1\frac{K'(x)}{\sqrt{(1+x)(1+2x)}}dx\overset{?}{=}2\int_0^1\frac{K'(\frac{1-x^2}{2})}{\sqrt{1+x^2}}dx $$

The right integral is linked to the Watson triple integral Prove the closed-form of $\int_{0}^{1}\frac{K^\prime\left (\frac{1-x^2}{2}\right)}{\sqrt{1+x^2}}\text{d}x$

$$ \int_0^1\frac{K'(\frac{1-x^2}{2})}{\sqrt{1+x^2}}dx= \frac23\int_0^1\frac{K(\frac{1+x^2}{2})}{\sqrt{1-x^2}}dx= \frac{2\sqrt2}{3\pi}\iiint\frac{dxdydz}{3-\cos x-\cos y-\cos z} $$ Where $xyz\in[0,\pi]^3$ , and $$ \iiint\frac{dxdydz}{3-\cos x-\cos y-\cos z}= \frac{\sqrt6}{96}\times\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right) $$ How do I get the following relation $$ \int_0^1\frac{K(x)}{\sqrt{3-x}}dx\overset{?}{=}\frac43\int_0^1\frac{K(\frac{1+x^2}{2})}{\sqrt{1-x^2}}dx $$ Does this mean $$ \int_{-1}^0\frac{K'(-x)}{\sqrt{(1-x)(1-2x)}}dx\overset{?}{=} \frac{4}{3i}\int_{\frac12}^{1}\frac{K(x)}{\sqrt{(1-x)(1-2x)}}dx $$ And one of my previous questions An integral about elliptic integral

Thanks (￣▽￣)ブ

Jay Hung
  • 191

1 Answers1

1

Since there are the ambiguties in the definition of the modulus $m=k^2$ in the literature, I use the orignal complete elliptic integral instead of $K$:

$$\int _0^1 \frac{K(k)}{\sqrt{(3-k) }} dk = \int _0^1\int _0^1\frac{1}{\sqrt{(3-k) \left(1-x^2\right) \left(1-k^2 x^2\right)}}dx\ dk$$

In change of variables in Mathematica

Inactive[Integrate][IntegrateChangeVariables[
     Inactive[Integrate][1/ Sqrt[(1 - x^2) (1 - k^2 x^2) (3 - k)], 
    {k, 0, 1} ], y,  y == 1 - k] , {x, 0, 1}]

$$\int _0^1\int _0^1\frac{1}{\sqrt{\left(1-x^2\right) (y+2) \left(1-x^2 (1-y)^2\right)}}\ dy \ dx$$

  \int _0^1\text{IntegrateChangeVariables}
   \left[\int _0^1\frac{1}{\sqrt{\left(1-x^2\right) \left(1-x^2 (y-1)^2\right) (y+2)}}dy,
   z,z=\frac{1-y}{2}\right]dx

$$\int _0^1\int _0^{\frac{1}{2}}\frac{2}{\sqrt{\left(1-x^2\right) (3-2 z) \left(1-4 x^2 z^2\right)}} \ dz \ dx$$

Numerics

  {NIntegrate[1/Sqrt[(1 - x^2) (1 - k^2 x^2) (3 - k)], 
  {k, 0, 1},  {x, 0, 1}],
 NIntegrate[1/Sqrt[(1 - x^2) (1 - x^2 (1 - k)^2) (2 + k)], 
    {x, 0, 1}, {k, 0, 1}],
   NIntegrate[2/Sqrt[(1 - x^2) (1 - 4 x^2 z^2) (3 - 2 z)], 
   {z, 0, 1/2}, {x, 0, 1}],
  1/(96 \[Pi] Sqrt[3])
     Gamma[1/24] Gamma[5/24] Gamma[7/24] Gamma[11/24] // N}

(1.17585, 1.17585, 1.17585, 1.17585})
Roland F
  • 5,122