We know that in 1748 Euler published the "Introductio in analysin infinitorum", in which, he released the discovery of the Euler's formula:
eix=cosx+isinx
But who was the first mathematician to convert this to the form we all know and love, the Euler's identity:
eiπ+1=0
When was this formula first explicitly written in this way?
Was it Bernoulli, Euler's teacher and mentor, or another more modern mathematician?
Answer
For a reference pointing to Euler, see :
discussing Bernoulli's thesis that l(−1)=l(+1)=0 :
le rayon [du cercle] est à la quatrieme partie de la circonference, comme √−1 à l√−1. Donc posant le rapport du diametre à la circonference =1:π, il sera 12π=l√−1√−1 et pertant l√−1=12π√−1.
In a nutshell, he derive for the area of the first quadrant of the unit circle the formula :
π4=14√−1l(−1),
from which : l(−1)=π√−1.
See also page 165-on where, starting from his formula "dont la vérité est suffissament prouvée ailleurs" :
x=cosϕ+√−1 sinϕ ,
posing C as the real logarithm of the positive quantity √(aa+bb)=c, he derives the general formula for the logarithm of :
a+b√−1=C+(ϕ+pπ)√−1.
With c=1 and C=0 he get :
l(cosϕ+√−1 sinϕ)=(ϕ+pπ)√−1.
Finally, with ϕ=0 [and thus : cosϕ=1 and sinϕ=0] :
l(+1)=pπ√(−1) and thus [for p=0] : l(+1)=0
and, with ϕ=π :
l(−1)=+−π√−1.
Euler in : Introductio in analysin infinitorum, Tomus Secundus (1748), Ch.XXI, page 290, uses i for an imaginary quantity :
Cum enim numerorum negativorum Logarithmi sint imaginarii (...) erit l(−n), quantitas imaginaria, quae sit =i.
But he does not say that the symbol i is such that i2=−1.
In the same Introductio, Tomus Primus, §138, the formula is written as :
ev√−1=cosv+√−1sinv.
In conclusion, Euler "knows" the identity and he is the "iventor" of i to name an imaginary quantity, but it seems that he never writed it in the "modern form", at least because he constantly writes √−1.
Note
See also Cuchy's Cours (1821) for Euler's identity ; again, √−1 is used.
I've not made an extensive research but, due to the fact that Cauchy uses systematically i for denoting an increment [see : Résumé des leçons sur le calcul infinitésimal (1823) ] :
Δx=i,
my conjecture is that we hardly find any use of i as imaginary.
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