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Partial Fractions
This post is based on a section called Integration by Partial Fractions in Morris Kline’s book on Calculus [1]. The first step in understanding partial fractions is learning about polynomials. Definition of a Polynomial From Wikipedia, the following statement defines a polynomial [2]. In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and…
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Short Overview of Special Relativity based on Zangwill
Here is a summary of Section 22.1 (Special Relativity) of Zangwill’s Electrodynamics textbook [1]. Basic Points of Special Relativity There are different observers. One observer moves with a constant velocity with respect to the other observer. A velocity vector has a direction & a magnitude, so a constant velocity means that the moving observer’s direction &…
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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…
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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…