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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).

The complex logarithm Using polar coordinates and Euler’s formula allows us to define the complex exponential as ex+iy = ex eiy (11) which can be reversed for any non-zero complex number written in polar form as ‰ei` by inspection: x = ln(‰); y = ` to which we can also add any integer multiplying 2… to y for another solution! 4.

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