Friday, 7 November 2014

complex numbers - Did Euler discover the Euler's identity



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 l1. Donc posant le rapport du diametre à la circonference =1:π, il sera 12π=l11 et pertant l1=12π1.




In a nutshell, he derive for the area of the first quadrant of the unit circle the formula :




π4=141l(1),





from which : l(1)=π1.






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+b1=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)=+π1.







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 :




ev1=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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