Final answer:
To evaluate the integral ∫ dz / z(e^z -1), we can use the Residue Theorem to find the residue at the simple pole z=0. The residue is found to be 1. By applying the Residue Theorem, we can evaluate the integral as 2πi.
Step-by-step explanation:
To evaluate the integral ∫ dz / z(e^z -1), we can use the Residue Theorem. The Residue Theorem states that for a function with a simple pole at a point c, the residue of the function at c is equal to the coefficient of the (z-c)^-1 term in the Laurent series expansion of the function around c. In this case, the function has a simple pole at z=0.
The residue at z=0 can be found by taking the limit as z approaches 0 of z times the function (z(e^z -1)) divided by e^z - 1. Simplifying this expression, we get the residue to be 1.
Since the unit circle is oriented counterclockwise, we can use the Residue Theorem to evaluate the integral as follows:
∫ dz / z(e^z -1) = 2πi * Res(z=0) = 2πi * 1 = 2πi