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If-x-R-show-that-2-i-e-1-3i-2-i-e-1-3i-is-also-real-




Question Number 37948 by Fawomath last updated on 19/Jun/18
If x ∈R  show that (2+i)e^((1+3i)) +(2−i)e^((1−3i))  is also real.
IfxRshowthat(2+i)e(1+3i)+(2i)e(13i)isalsoreal.
Commented by prof Abdo imad last updated on 19/Jun/18
let prove first that conj( e^(a+ib) ) =e^(a−ib)   if a and b reals  (conj(z)=z^− )  we have e^(a +ib) =e^a ( cosb  +i sinb )⇒  vonj(e^(a+ib) ) =e^a (cosb −isinb)=e^(a−ib)   let  z =(2+i) e^(1+3i)  ⇒z^− =(2−i)e^(1−3i)  ⇒  (2+i)e^(1+3i)  +(2−i)e^(1−3i)  =z +z^− =2Re(z) ∈ R.
letprovefirstthatconj(ea+ib)=eaibifaandbreals(conj(z)=z)wehaveea+ib=ea(cosb+isinb)vonj(ea+ib)=ea(cosbisinb)=eaibletz=(2+i)e1+3iz=(2i)e13i(2+i)e1+3i+(2i)e13i=z+z=2Re(z)R.

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