Essays/Euler's Identity
Euler's identity
0 = 1 + ^ 1p1 * 0j1
is the most beautiful equation in all of mathematics, relating in one short phrase the fundamental quantities 0, 1, {\displaystyle e}, {\displaystyle \pi }, and 0j1 and the basic operations plus, times, and exponentiation. It is a particular case of Euler's formula
(^0j1*x) = (cos x) + 0j1 * (sin x)
Standing on the shoulders of giants, herewith an informal proof of Euler's formula and Euler's identity.
The adverb t. is such that f t. i gives the i-th coefficient of the Taylor expansion of function f . Thus:
^ t. i.10 1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413 2.48016e_5 2.75573e_6
If the right argument of the derived function is extended precision, then so too is the result:
^ t. i.10x 1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880
Similarly for sin and cos:
sin=: 1&o. cos=: 2&o. sin t. i.10x 0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 cos t. i.10x 1 0 _1r2 0 1r24 0 _1r720 0 1r40320 0
The related adverb t: gives the coefficients weighted by %!i . Thus:
] e=: ^ t: i.10 1 1 1 1 1 1 1 1 1 1 ] s=: sin t: i.10 0 1 0 _1 0 1 0 _1 0 1 ] c=: cos t: i.10 1 0 _1 0 1 0 _1 0 1 0
Now for ^0j1*x , the coefficients of the weighted series are:
] e1=. e * 0j1 ^ i.10 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1
Aligning the columns of e1 and the coefficients for sin x and cos x , we get:
e1 , s ,: c 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1 0 1 0 _1 0 1 0 _1 0 1 1 0 _1 0 1 0 _1 0 1 0
Staring at this for a while, we note that on terms 0 2 4 6 8 ... e1 is the same as for c , and on terms 1 3 5 7 ... e1 is s*0j1 :
e1 , (s * 0j1) ,: c 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1 0 0j1 0 0j_1 0 0j1 0 0j_1 0 0j1 1 0 _1 0 1 0 _1 0 1 0
That is:
e1 ,: (s * 0j1) + c 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1 e1 = (s * 0j1) + c 1 1 1 1 1 1 1 1 1 1
In other words,
(^0j1*x) = (cos x) + 0j1 * sin x
Since j.y is 0j1*y and x j. y is x+0j1*y ,
(^j.x) = (cos x) j. (sin x)
It is not too far a leap to plug in extremal values of the functions. Thus:
pi=: o. 1 cos pi _1 sin pi 1.22461e_16 ^ 0j1 * pi _1j1.22461e_16
In J7.01, the composition ^@o. ({\displaystyle e^{z}} composed with {\displaystyle \pi } times) is supported by special code, and
^@o. 0j1 _1
I used to think math was no fun,
'Cause I couldn't see how it was done.
Now Euler's my hero,
For I now see why 0
Equals e to the π i plus 1.
e raised to the π times i,
And plus 1 leaves you nought but a sigh,
This fact amazed Euler,
That genius toiler,
And still gives us pause, bye the bye.
—— Anon y Mous
See also
Contributed by Roger Hui.[[Category:LeonhardEuler]