Euler’s identity and Ogawa’s Professor’ beloved equation
Today is π day so, let’s celebrate it by deriving one of the most beautiful relationships in mathematics: Euler’s identity (the Professor’s beloved equation).
Some -smooth and analytic- functions f are such that:
Where T is the funcion’s Taylor expansion. If c = 0, then we call it f’s Maclaurin series:
Thus, the Maclaurin series for f(x) = exp(x) is given by:
Now, if we let x = iθ , then:
Since the Maclaurin series for the sine and cosine functions are:
And,
Then,
For θ = π, we get:
Hence,
Which is the Professor’s beloved equation. The equation written by the Professor to end the argument between his sister-in-law and the houskeeper.
Yōko Ogawa’s “The Professor and the Houskeeper” is a heartwarming novel about an elderly number theorist whose memory lasts for only 80 minutes. After that lapse, he cannot remember anything but the memories he accumulated before the car accident.
This is not a story about mathematics. Of course, the love for number theory is inherent and omnipresent but, the beauty of the bond between the housekeeper, the Professor and the housekeeper’s son (called Root, by the Professor) exceeds every other element within the story.
“He treated Root exactly as he treated prime numbers. For him, primes were the base on which all other natural numbers relied; and children were the foundation of everything worthwhile in the adult world.”
The Professor, who despite his condition, never abandoned his mathematics, transmitted his love for discrete mathematics to both, the Houskeeper and Root. She -particularly- discovered the beauty of mathematics. In exchange, she granted him with true, uninterested friendship.
And, that’s all what this story is about: friendship, fraternal love, tenderness. About the joy of experiencing the happiness of our loved ones. About admiration, respect. About the virtue of never abandoning.
This book is an absolute must! Your heart will definitely melt.
Happy pi day!
