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

Question Number 152203 by peter frank last updated on 26/Aug/21

If x is real show that

{(2 + i)}^{(1 + 3 i) x} + {(2 - i)}^{(1 - 3 i) x}

is also real

Commented by MJS_new last updated on 26/Aug/21

{(e^{θ i} r)}^{a + b i} + {(e^{- θ i} r)}^{a - b i} =

= e^{- θ b + θ a i} r^{a + b i} + e^{- θ b - θ a i} r^{a - b i} =

= (e^{θ a i} r^{b i}) (e^{- θ b} r^{a}) + (e^{- θ a i} r^{- b i}) (e^{- θ b} r^{a})

easy to see the shape of these is

e^{i ϕ} s + e^{- i ϕ} s = 2 s \cos ϕ

which obviously is real

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