Transcription of EULER’S FORMULA FOR COMPLEX EXPONENTIALS
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EULER’S FORMULA FOR COMPLEX EXPONENTIALS
euler S FORMULA FOR COMPLEX EXPONENTIALSA ccording to euler , we should regard the COMPLEX exponentialeitas related tothe trigonometric functionscos(t)andsin(t)via the following inspired definition:ei t= cost+isintwhere as usual in COMPLEX numbersi2= 1.(1)The justification of this notation is based on the formal derivative of both sides,namelyddt(ei t)=i(ei t) =icost+i2sint=icost sintsincei2= 1ddt(cost+isint) = sint+icostsinceiis a with the initial value of 1 for both sides att= 0, assuminge0= 1holds forcomplex values motivation for looking at this combination comes from the link between pointin the plane with coordinates(x, y)and COMPLEX numbers formed by the relationz=x+iy, sincezbecomes the combinationrcos +irsin , which suggeststhat the combination may be interesting to look at (unit circle hasr= 1).This turns out to be a very important unification and simplification of many resultsin both trigonometry and calculus, in which the FORMULA leads us to correct manip-ulations.
EULER’S FORMULA FOR COMPLEX EXPONENTIALS According to Euler, we should regard the complex exponential eit as related to the trigonometric functions cos(t) and sin(t) via the following inspired definition:eit = cos t+i sin t where as usual in complex numbers i2 = ¡1: (1) The justification of this notation is based on the formal derivative of both sides,
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