Consider this example

>>> limit(elliptic_k(2+I*x), x, 0, "+")
elliptic_k(2)
>>> limit(elliptic_k(2+I*x), x, 0, "-")
elliptic_k(2)

As can be seen through wolfram

We are missing something i.e. ((2 I) Sqrt[Pi] Gamma[5/4])/Gamma[3/4]