Question: how to solve complex equations?

aab := int(exp(-a*x)*cos(b*x), x=0..infinity);
simplify(eval(aab));
 

why above equation is not equal to a/(a^2+b^2) ?

 

and how to solve below equation

z assuming complex;
solve(z^n - z = 0, z);

it only return 1 or 0, but not complex equations in terms of n

 

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