Write the elliptic integral \(K_{1}(\kappa)\) in the form

\[
K_{1}(\kappa)=\int_{0}^{\pi / 2} \frac{1-\kappa \sin \phi}{\sqrt{ }\left(1-\kappa^{2} \sin ^{2} \phi\right)} d \phi+\int_{0}^{\pi / 2} \frac{\kappa \sin \phi}{\sqrt{ }\left(1-\kappa^{2} \sin ^{2} \phi\right)} d \phi
\]

and show that, as \(\kappa \rightarrow 1-\), the first integral \(\rightarrow \ln 2\) while the second \(\approx \ln \left[2 /\left(1-\kappa^{2}\right)^{1 / 2}\right]\). Hence \(K_{1}(\kappa) \approx \ln \left(4 /\left|\kappa^{\prime}\right|\right)\), as in equation (13.4.34).