- #1
Bashyboy
- 1,421
- 5
Homework Statement
Let ##C## be the negatively oriented contour ##2e^{-i \pi t}## ##0 \le t \le 1## For which of
the following functions does the Cauchy-Goursat theorem apply so that the integral of the
function along ##C## is zero?
Homework Equations
The Attempt at a Solution
The first function is ##f(z) = e^{z^7}##, where ##z^7 = e^{7 \log z}##.
Obviously, the contour over which we wish to integrate ##f(z)## is the unit circle. For ##f(z)## to even be integrable, it must first be a function, which can be accomplished by removing a branch so that ##\log z## becomes a function. By removing a branch, we are actually removing a ray which makes some angle ##\theta## with the positive real-axis that begins at the origin and extends away from it. For instance, I could remove the ray which corresponds to the negative real-axis, all those points of the form ##(a,0)##, where ##a \in \mathbb{R} \setminus \{0\}##; consequently, there can be no points whose angle (argument) is ##\theta = \pi##.
Here is why I do not believe one can apply the Cauchy-Goursat theorem: there are singularities interior to the contour (infinitely many, actually), and one singularity on the contour--namely, ##(-1,0)##.
Despite this, I believe we can integrate the function over the contour because there is only one singularity point. This would be analogous to an improper integral because we are integrating up to a singularity point, right?
My question is, how would I set up the integral? When ##t = 1/2##, we get ##C(1/2) = (-1,0)##. So, would I do an improper integral on the variable ##t##?
Also, would the integral be the same independent of which branch I choose to remove?