De Moivre's theorem for rational index

If n is the rational number , then the value of or one of the values of \[(cos(\theta )+isin\theta )^n\] is \[ cos(n\theta)+isin(n\theta ). \]

In fact ,if n=\[ \frac{p}{q}\] where p and q are integers , q > 0 and p ,q have no factor in comman. Then\[(cos(\theta )+isin\theta )^n\] has q distinct values , one of which is \[ cos(n\theta)+isin(n\theta )\]

Boundary condtion - \[ k=0,1,2,..........................,q-1\]

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