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Complementary complete elliptic integral of the first kind ( $K^{'} (k)$ )

For the complementary modulus

k^{'} = 1 - k^{2}

,

K^{'} (k) = K (k^{'}) = \int_{1}^{1/ k} \frac{d t}{( t ^{2} - 1 ) ( 1 - k ^{2} t ^{2} )} .

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Solution