Christina DanielA Definite Integral of the Complex Exponential Function
Recall from this post that:
$ \int_0^{2 \pi} d\theta \cos\theta + i \int_0^{2 \pi} d\theta \sin\theta = 0$
and
$ \int_0^{2 \pi} d\theta e^{i \theta} = 0.$
These are definite integrals because the upper and lower limits of integration are finite numbers.
Symmetry of the Sine and Cosine Functions
Due to the symmetry of the sine and cosine functions, it is possible to add another $2 \pi$ radians to the upper limit of each integral, to get:
$ \int_0^{4 \pi} d\theta \cos\theta + i \int_0^{4 \pi} d\theta \sin\theta = 0$
and
$ \int_0^{4 \pi} d\theta e^{i \theta} = 0.$
Introducing Negative Angular Values
Thus far, only positive and zero values of an angle have been considered. It is possible to establish a meaning for negative angular values in radians. While a counterclockwise angular displacement corresponds by convention or definition to a positive angle, it is logical to associate clockwise angular displacement with a negative angle. Hence a negative angular value–in radians or degrees or any other angular unit–corresponds to moving in a clockwise manner from 0 radians on the positive $x$-axis to some other nonzero angular value.
By symmetry of the sine and cosine functions, and using the plotting reasoning in this post, the lower limits of integration can adjusted as follows:
$ \int_{-2 \pi}^{2 \pi} d\theta \cos\theta + i \int_{-2 \pi}^{2 \pi} d\theta \sin\theta = 0$
and
$ \int_{-2 \pi}^{2 \pi} d\theta e^{i \theta} = 0.$
Adjusting the Upper Limit of Integration
In fact, by the symmetry of the cosine and sine functions, the upper limits of integration can be an arbitrary, finite integer $N$ multiple of $2 \pi$ radians:
$ \int_{-2 \pi}^{2 \pi N} d\theta \cos\theta + i \int_{-2 \pi}^{2 \pi N} d\theta \sin\theta = 0$
and
$ \int_{-2 \pi}^{2 \pi N} d\theta e^{i \theta} = 0.$
Adjusting the Lower Limit of Integration
By symmetry of the sine and cosine functions, the same can be said about the lower limits of integration, for which a finite integer $M$ is introduced:
$ \int_{-2 \pi M}^{2 \pi N} d\theta \cos\theta + i \int_{-2 \pi M}^{2 \pi N} d\theta \sin\theta = 0$
and
$ \int_{-2 \pi M}^{2 \pi N} d\theta e^{i \theta} = 0.$
Two integers $M$ and $N$ are used so that there is flexibility in defining the values of the upper and lower limits of integration. In particular, the upper limit need not be the same as the lower limit of integration! This conclusion follows from the symmetry of the sine and cosine functions–the values of each trigonometric function can be imagined on a plot versus the angular variable, $\theta$.