Find the order of each of the zeros of z2 4z 42 b Locate a

Find the order of each of the zeros of (z^2 - 4z + 4)^2 b) Locate and classify the isolated singularities of e^8 - 1/e^7 - 1.

Solution

(a) ( z² - 4z + 4)² = ( z - 2)⁴ There is only one zero of this expression, which is + 2. This zero is of the multiplicity/order 4.

(b) The expression is not clear. It is not clear whether the power of e in the numerator/denominator is z or 7.Assuming that in both the cases, it is z, the sinularity of the function ( e^z - 1)/ (e^z - 1) is where e^z - 1 = 0 i.e. where e^z = 1. On taking natural logarithm of both the sides, we have z ln e = ln 1 or, z = 0 ( as ln e = 1 and ln 1 = 0) . Now, z = 0 is an isolated sinularity as there is no other singularity in the nearby region.