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At $z=1$, the branch value is the complete elliptic integral of the first kindA contour that goes once around $1$ and then to $z$ applies the reflection about $K$, giving

Similarly, the branch value at $-1$ is $-K$, so a loop around $-1$ acts as $R_{-1}(w)=-2K-w$. A contour that loops first around $-1$ and then around $1$ therefore gives

Finally approach the interval $(1,1/k)$ through the upper half-plane. Thereso the branch value at $1/k$ isIt follows that a loop around $1/k$ acts asA contour that loops first around $1$ and then around $1/k$ consequently givesHere $L=K^{\prime }(k)$ is the complementary complete elliptic integral of the first kind. These contours exhibit all three requested values.