Articles

Physics is motivated by observed phenomena and seeks to explain the empirical data using fundamental laws as first principles. These articles seek to lay out some of the basic definitions and applications we use, focusing on the math to support the laws, in order to get started with this vibrant field.

An introduction to these articles.
An end to the mathematical frustration of man comes with analysis of the binomial (1620), and the generalization of the formula to a series solution of the binomials with non-natural number exponent (1665).
If the summation of a series grows faster than some rate then it diverges. Up until the 17th century, no one knew whether the Harmonic series diverged, or converged, as it grows very slowly, but faster than the Geometric Series.
Exponents are a simple algebraic construction and notation; they are used in scientific notation and are implicit in the decimal notation. Roots are an intrinsic part of exponents (the fractional exponents). So, in this section the original exponent value (zero) is analyzed.
The logarithm is presented as an 18th century revelation. The natural exponential is compared to other exponential bases, graphically.
Tangents in general are considered, particularly those of curves which come to the minds of early modern natural philosophers.
The inverse operation of calculating the tangents of a curve, or derivative, is the area under the curve.
The representation of a curve with a differential series describes its local composition.
Trigonometry is the study of the Sine and Cosine cofunctions, their domains and ranges, along with the rational combination of the two as Tangent, with the Taylor Series derived.
Here, the Trigonometric Taylor Series' are juxtaposed with the Exponential Series of an imaginary argument to derive, Euler's Formula.
A theorem of Mertzbacher is presented, using Fourier analysis to outline a direct route from wave-particle duality to their complex existence.
Bibliographical references.