Deriving Euler's Formula via Algebra and Basic Calculus

Leonhard Euler's formula, a cornerstone in the field of mathematics, embodies a profound relationship between trigonometric functions and complex exponential functions. It is typically attributed to both algebraic and calculus techniques, particularly the Taylor series expansion. However, there are instances where Euler's formula can be derived using just algebra and basic calculus principles. This article explores this intriguing derivation, following the steps laid out by Richard P. Feynman in his Feynman Lectures on Physics.

Introduction to Euler's Formula

Euler's formula is expressed as:

[ e^{ix} cos(x) isin(x) ]

This beautifully simple equation connects the exponential function with trigonometric functions through the imaginary unit (i). To derive this formula, we will follow the path taken by Feynman, which emphasizes the underlying algebraic structure without the need for advanced calculus techniques.

Derivation Using Taylor Series

The key to this derivation lies in the Taylor series expansions for the exponential function and the trigonometric functions (cos(x)) and (sin(x)). The Taylor series for (e^{ix}) is:

[ e^{ix} 1 ix frac{(ix)^2}{2!} frac{(ix)^3}{3!} frac{(ix)^4}{4!} cdots ]

Expanding and reorganizing the terms, we get:

[ e^{ix} 1 - frac{x^2}{2!} frac{x^4}{4!} - frac{x^6}{6!} cdots ileft(x - frac{x^3}{3!} frac{x^5}{5!} - frac{x^7}{7!} cdotsright) ]

The series can be split into two parts: one containing only even powers of (x) and the other containing only odd powers of (x). Recognizing these as the Taylor series expansions for (cos(x)) and (sin(x)), respectively, we can rewrite the formula as:

[ e^{ix} cos(x) isin(x) ]

This completes the derivation using basic algebra and the Taylor series, making it clear and accessible without any need for advanced calculus techniques.

Historical Context and Log Tables

Another interesting historical context emerges when considering how log tables were created before the advent of electronic computers. According to Feynman, the creation of log tables was accomplished by leveraging only 10 square roots and some multiplication. This method, while not directly related to deriving Euler's formula, provides a fascinating insight into the power of basic mathematical operations and their practical applications.

Conclusion

James Clerk Maxwell's first step in his derivation of the Euler formula also reveals simple elegance in mathematics. By looking at the Taylor series of (e^{ix}), we can derive Euler's formula with just basic algebraic manipulation. This approach not only simplifies the derivation but also underscores the beauty and interconnectedness of mathematical concepts.

For those interested in delving deeper into this topic, you may explore Feynman's detailed explanation in his lectures. This resource offers a rich and accessible exploration of the subject, encouraging a deeper understanding and appreciation of mathematical elegance.