• Multiple Cycles of the Complex Exponential Function

    A 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…

  • One Cycle of a Trigonometric Function

    Integral of a Complex Exponential Consider the integral $ \int_0^{2 \pi} d\theta e^{i \theta} $. Upper Limit of $2 \pi$ Radians The upper limit is the number, $2 \pi$. For a circle with radius $r=1$, the circumference of the circle is $2 \pi r = 2 \pi$. Even though $2 \pi$ is a number, it…