I'm having some difficulties understanding when I'm supposed to take branch cuts into consideration when facing contour integrals. So far in my textbook, the exercises will always point out whether we are working with branches or not. But does that mean that I only have to consider branch cuts when the exercise specifically points it out?
To illustrate, here is one example given in the book:
The function f(z) = 1/(z^2), which is continuous everywhere except at the origin, has an antiderivative F(z) = -1/z in the domain |z| > 0, consisting of the entire plane with the origin deleted. Consequently,
INTEGRAL (1/(z^2))dz = 0
when C is the positively oriented circle:
z = 2e^(iθ)
(- pi < θ < 2 pi)
And here the example ends.
Later in the same section I run into this example:
Let us use an antiderivative to evaluate the integral
where the integrand is the branch
f(z) = z^(1/2) = exp((1/2)log(z)) = r^(1/2)*e^(iθ/2)
(r > 0, 0 < θ < 2 pi )
of the square root function and where C is any contour from z = -3 to z = 3 that, except for its end points, lies above the x-axis.
The example then goes on to explain that the branc of z^(1/2) is not defined at θ = 0, and that thus we have to establish a new branch before we can proceed to solve the integral.
I don't have any problems performing this latter process. However, my question is, how do we know that we have to take branch cuts into consideration in example 2 but not in example 1? Is it just because it is specifically pointed out in the example that we now work with an initial branch? Does this mean that if there is no such initial branch specified in an exercise, I can just solve away without taking it into consideration? (As long as the integral itself does not involve a Log of course).
I would greatly appreciate it if anyone can explain this to me!