Sunday, December 6, 2015

On the Contributions of Leonard Euler

After reading Euler: The Master of Us All and excerpts from Journey Through Genius, I have felt inclined to speak about the incredible mathematician and person, Leonard Euler.

The sheer volume of Euler's work cannot be appreciated, nor comprehended until one gazes at how many shelves the volumes of his life work takes up. The quality and influence of Euler's work is not grasped until one studies many areas in math and realizes how many groundbreaking discoveries bear Euler's name (and many he worked that don't.) The saying goes: "...there is ample precedent for naming laws and theorems for persons other than their discoveries, else half of analysis would be named for Euler."

Euler's creativity withing mathematics is astounding. Take for example his proof to solve for the infinitie sum 1+1/4+1/9+1/16+... where the denominators are all squares. He starts by introducing a infinite sum and showing that it's equal to sin(x)/x. He then solves for the zeros of sin(x), changes the sin(x)/x into root form, and then equates this back to his original sum, getting two infinite sums to equal each other. He then separates terms by their power (x^n) and equates infinite sums to these finite coefficients. This yields not just the sum of squares, but the sum of any even power: an incredible result!.

Never has there been a mathematician as prolific as Leonard Euler, and I daresay there will never be again. Although this is not because someone as talented and hardworking will never exist; they surely will! Euler also had luck on his side: the time period Euler lived still had many discoveries left open. Not that we don't have unanswered questions today but they are seemingly getting more and more difficult to solve. As my advisor Shane Larson tells me: "There is less and less 'low hanging fruit'." None of this is to degrade his achievements; far from it. Just the thought that we could again have someone so prolific is exciting but I just doubt that a modern mathematician could be able to keep up with the inane pace of Euler.

But I digress. Euler will continue to be a revered thinker. His works will be studied for centuries, as there is much any young aspiring mathematician can learn from style, work ethic, and simple curiosity. Truly, Euler is the master.

1 comment:

  1. Always love to see Euler appreciation. For content/complete, it would be nice to add a few of the details in paragraph 3 if they're available. What was the infinite series? What is the sum of the inverse squares? Is the generalized sum for even powers a nice closed form? This is a great example, as it shows his creativity and use of one of his favorite tools.
    Other Cs +