If x ∈R show that (2+i)e^((1+3i)) +(2−i)e^((1−3i)) is also real.


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Question Number 37948 by Fawomath last updated on 19/Jun/18

$I f x \in R$ $s h o w t h a t (2 + i) e^{(1 + 3 i)} + (2 - i) e^{(1 - 3 i)} i s a l s o r e a l .$
Commented by prof Abdo imad last updated on 19/Jun/18

$l e t p r o v e f i r s t t h a t c o n j (e^{a + i b}) = e^{a - i b}$ $i f a a n d b r e a l s (c o n j (z) = \bar{z})$ $w e h a v e e^{a + i b} = e^{a} (c o s b + i s i n b) \Rightarrow$ $v o n j (e^{a + i b}) = e^{a} (c o s b - i s i n b) = e^{a - i b} l e t$ $z = (2 + i) e^{1 + 3 i} \Rightarrow \bar{z} = (2 - i) e^{1 - 3 i} \Rightarrow$ $(2 + i) e^{1 + 3 i} + (2 - i) e^{1 - 3 i} = z + \bar{z} = 2 R e (z) \in R .$
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