Scientific Space once mentioned the formula e^{i\pi}+1=0, which is hailed as one of the "most remarkable formulas in mathematics." Readers might have heard of or even proven it long ago. However, do you know other anecdotes about e, i, \pi? For example, do you know what i^i equals? Or i^{1/i}?
This article invites you to appreciate the beauty of mathematics!
In 1719, the Italian amateur mathematician Fagnano (1682–1766) obtained: e^{\pi/4} = \left[\frac{1-i}{1+i}\right]^{i/2} This is yet another formula that links the three together!
One might assume that i^i is an imaginary number, but the fact is surprising. Euler derived from the above formula that it is a real number, and it also connects e, i, \pi: i^i = e^{-\pi/2} From this, we also have i \ln i = -\frac{\pi}{2}. Of course, this is actually a multi-valued function.
Like e^{i\pi}+1=0, this formula can be considered a perfect crystallization! Similarly, there is: i^{1/i} = e^{\pi/2} This expression is easily derived from the previous one.
An interesting fact about e^{\pm ix} = \cos x \pm i \sin x is that the legendary Indian mathematician Ramanujan independently derived it at the age of 12.
The April 1975 issue of Scientific American published a "mathematical joke": e^{\pi\sqrt{163}} = 262,537,412,640,768,744. Note that the right side is an integer! However, since it was called an international joke, it is not actually an integer; the right side should be equal to: 262,537,412,640,768,743.999999999999250\dots It is said that Ramanujan was also the first to conjecture that the right side "should" be an integer.
There is also an infinite series where e and \pi "share the same room": \frac{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} + \dots + \frac{1}{1 + \frac{1}{1 + \frac{2}{1 + \frac{3}{1 + \frac{4}{1 + \dots}}}}}
(Content sourced from The Incredible e)
There is a famous formula used to approximate factorials called Stirling’s formula: n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n. it also "incorporates" e and \pi.
In mathematics, there is an improper integral (Gaussian integral): \int_{-\infty}^{+\infty} e^{-x^2} dx = \sqrt{\pi}
These formulas and theorems all ingeniously and unexpectedly link e, i, \pi together, leaving us in awe!
With my limited knowledge, it is difficult for me to vividly present the beauty of mathematics to everyone. I can only mention it briefly here, hoping to serve as a starting point for further discussion ^_^.
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