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Shahn Majid and Eliezer Batista find the derivative of a wave plane in their paper:

Non Commutative Geometry of Angular Momentum Space $U \mathfrak{su}(2)$.

They obtained the non commutating relations between one form:

$x_a dx_b=(dx_b)x_a+\frac{i \lambda}{2} \epsilon_{abc} dx_c+\frac{\lambda}{4} \delta_{ab} \theta$

$x_a \theta=\theta x_a +\lambda dx_a$

They also find that:

$df(x)=(dx_a)\partial^a f(x)+\frac{\theta}{c} \partial^0 f(x)$

How can they deduce that:

$de^{ik.x}=\left\{\frac{\theta}{\lambda}\left(\cos \left(\frac{\lambda \vert k \vert}{2}\right)-1\right)+\frac{2i \sin\left(\frac{\lambda \vert k \vert}{2}\right)}{\lambda \vert k \vert} k. dx \right\} e^{ik.x}\quad?$

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    Arm Boris: I edited your question, to enlarge the final formula about which you ask. I want to be sure I didn't inadvertently lose or change any terms2012-11-27

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