I have an integral depending on the following expression $$-(k_{20}-k_{10})^2 + (k_{2x}-k_{1x})^2 + (K_{2z}-K_1)^2 - K_2^2$$ where the integration variables are $k_{20}$ and $K_{2z}$. I need transformation (possibly a "rotation", even if it is, strickly speaking, a boost) such that this expression becomes $$(k_{20}'-\sqrt{k_{10}^2-K_1^2})^2 + (k_{2x}-k_{1x})^2 + K_{2z}'^2 - K_2^2.$$ Now if $k_{10}^2-K_1^2>0$ the transformation is easy to find and it is $$k_{20}' = \frac{\sqrt{k_{10}^2-K_1^2}}{k_{10}^2-K_1^2}k_{10}k_{20} - \frac{\sqrt{k_{10}^2-K_1^2}}{k_{10}^2-K_1^2}K_1K_{2z}$$ $$K_{2z}' = -\frac{\sqrt{k_{10}^2-K_1^2}}{k_{10}^2-K_1^2}K_1k_{20} + \frac{\sqrt{k_{10}^2-K_1^2}}{k_{10}^2-K_1^2}k_{10}K_{2z}.$$ The problems occur when $k_{10}^2-K_1^2<0$. I can only reconduce the expression to $$(k_{20}'-i\sqrt{k_{10}^2-K_1^2})^2 + (k_{2x}-k_{1x})^2 + K_{2z}'^2 - K_2^2$$ and one of the factors of the change of variable is imaginary. My questions are:
Is it possible to find a transformation for $k_{10}^2-K_1^2<0$ to obtain $(k_{20}'-\sqrt{k_{10}^2-K_1^2})^2 + (k_{2x}-k_{1x})^2 + K_{2z}'^2 - K_2^2$ without using complex factors?
If it is not possible, when is it legitimate to change the integration variable from real to complex? I can't find anything online or in my textbooks.