Calculation of $\int _ { \gamma } z ^ { i} d z$, where $\gamma$ is any path (contour) from $- 1$ to $1$ lying above the $x$-axis.

Question

How can I calculate $\int _ { \gamma } z ^ { i} d z$, where $\gamma$ is any path (contour) from $- 1$ to $1$ lying abovethe $x$-axis. Can I use $$z ^ {i } = e ^ { i \log z }$$ on $$z \in D = \left\{ \begin{array} { l } { z \in \mathbb { C }: | z | > 0 }, \\ { - \pi / 2 < \text { Arg } z < \frac { 3 \pi } { 2 } } \end{array} \right.$$ to get the integral as the difference of the values of anti-derivatives at end points of $\gamma$.

Answer 1

Since the upper half-plane is simply connected and $f(z)=z^i$ is analytic in the interior of this set, Cauchy's integral theorem guarantees that the value of an integral between any two points in the set will be path-independent. These criteria also guarantee the existence of a complex antiderivative (see Wikipedia).

The easiest way to proceed from here is to calculate an antiderivative and calculate an antiderivative $g(z)$ and calculate the integral as follows: $$\int_\gamma z^i\ dz=g(1)-g(-1)$$ In this case, an antiderivative is not too difficult to obtain: $$g(z)=\frac{z^{i+1}}{i+1}$$

Another method is to make use of the path-independence. Since the value will be the same no matter the choice of $\gamma$, we can simply pick any contour in the upper half-plane and calculate the integral. An easy choice in this case is the circle $|z|=1$ running from $-1$ to $1$.