Some Little-Known Facts About e, i, π...
Scientific Space has previously mentioned $e^{i\pi}+1=0$, that formula hailed as "one of the most remarkable formulas in mathematics." Readers probably heard about it, or even proved it, long ago. But do you know some of the other curious anecdotes involving e, i, and π? For instance, do you know what $i^i$ equals? Or $i^{1//i}$?
Let's take this opportunity to appreciate the beauty of mathematics! more
In 1719, the Italian amateur mathematician Fagnano (1682–1766) obtained:
$$e^{\pi//4}=[\frac{1-i}{1+i}]^{i//2}$$
Here again is a formula linking all three constants together!
You might well assume that $i^i$ is an imaginary number, but the truth is rather surprising. From the equation above, Euler found that it is in fact a real number—and one that ties e, i, and π together:
$$i^i=e^{-\pi//2}$$
From this we also get $i ln i=-\frac{\pi}{2}$. Of course, strictly speaking this is a multivalued function.
Just like $e^{i\pi}+1=0$, this formula is a crystallization of perfection! There's a similar one:
$$i^{1//i}=e^{\pi//2}$$
which can easily be derived from the previous formula.
An interesting fact about $e^{+- ix}=cos x +- i sin x$ is that the legendary Indian mathematician Ramanujan independently derived it at the age of 12.
In April 1975, Scientific American published a "mathematical joke": $e^{\pi\sqrt{163}}=262 537 412 640 768 744$, noting that the right-hand side is supposedly an integer! But since it was announced as an April Fools' joke, it is in fact not an integer—the right-hand side actually equals:
262537412640768743.999999999999250......
It's said that Ramanujan was also the first to conjecture that the right-hand side "should" be an integer.
There's also an infinite series where e and π "share a room":
$$1/2 \sqrt{e\pi}=1+\frac{1}{1\cdot 3}+\frac{1}{1\cdot 3\cdot 5}+\frac{1}{1\cdot 3\cdot 5\cdot 7}+...+\frac{1}{1+\frac{1}{1+\frac{2}{1+\frac{3}{1+\frac{4}{1+...}}}}}$$
(Content sourced from The Incredible e.)
There's the famous Stirling's formula for approximating factorials: $n! \approx \sqrt{2\pi n} (\frac{n}{e})^n$, which likewise brings e and π together under one roof.
And there is a well-known improper integral (the Gaussian integral) in mathematics: $\int_{-\infty}^{+\infty} e^{-x^2} dx=\sqrt{\pi}$
All these formulas and theorems tie e, i, and π together in ways that are both ingenious and unexpected—truly a sight to marvel at!
Given my limited knowledge, I can't do full justice to the beauty of mathematics here—I can only offer this brief sketch, hoping it might inspire further exploration. ^_^
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.