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).
posite numbers), a fast algorithm exists to compute these (fast FT). 3. Calculus: The functions of the form eat cos bt and eat sin bt come up in applications often. To find their derivatives, we can either use the product rule or use Euler’s formula (d dt)(eat cos bt+ieat sin bt) = (d dt)e(a+ib)t = (a+ib)e(a+ib)t = (a+ib)(eat cos bt+ieat sin ...
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