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You can also use Refine with Element : Refine[Sqrt[2] Conjugate[Sqrt[1/L]] Sin[(Pi* Conjugate[n x])/Conjugate[L]], {Element[L, Reals], Element[n, Integers]}] gives and if you add that L>0: Refine[Sqrt[2] Conjugate[Sqrt[1/L]] Sin[(Pi* Conjugate[n x])/Conjugate[L]], {Element[L, Reals], Element[n, Integers], L > 0}] Other simple examples : 1. ...


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Conjugate by default assumes that all symbolic quantities are potentially complex. This may seem annoying at first, but there is a very good reason for it, and one way to see why is to define your own version of Conjugate, and see it fail. For educational purposes, I do that below. Define $Conjugate as follows: $Conjugate[x_] := x /. Complex[a_, b_] :> ...


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1. See the function ComplexExpand ComplexExpand[Re[x + I y]] gives x 2. Also you can try Refine combined with Element: Refine[Re[x + I y], Element[x, Reals]] (*x - Im[y]*)


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First we can see that ${\frac{(6k+1)^{k}}{(2k+5)^{k}}}$ behaves asymptotically as $3^k$, while $(z-2i)^k$ is divergent if $\|z-2i\|>1 $, however when $\|z-2i\|<1 $ it is convergent. For $\|z-2i\|=1$ this criterion is not conclusive. On the other hand we can carefully extend this argument to the full sequence, so we need only $\|z-2i\| < ...



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