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simplify-w-n-1-in-n-1-in-n-with-n-integr-natural-




Question Number 66032 by mathmax by abdo last updated on 07/Aug/19
simplify  w_n =(1+in)^n −(1−in)^n  with n integr natural
simplifywn=(1+in)n(1in)nwithnintegrnatural
Commented by mr W last updated on 08/Aug/19
=i 2(1+n^2 )^(n/2) sin (n tan^(−1) n)
=i2(1+n2)n2sin(ntan1n)
Commented by mathmax by abdo last updated on 08/Aug/19
we have 1+in =(√(1+n^2 ))e^(iarctann)  ⇒(1+in)^n =(1+n^2 )^(n/2)  e^(inarctan(n))   also 1−in =vonj(1+in) =(√(1+n^2 ))e^(−iarctan(n))   ⇒  (1−in)^n  =(1+n^2 )^(n/2) e^(−inarctan)n))  ⇒  w_n =(1+n^2 )^(n/2) {2i sin(narctan(n)} =2i (1+n^2 )^(n/2)  sin(narctan(n))
wehave1+in=1+n2eiarctann(1+in)n=(1+n2)n2einarctan(n)also1in=vonj(1+in)=1+n2eiarctan(n)(1in)n=(1+n2)n2einarctan)n)wn=(1+n2)n2{2isin(narctan(n)}=2i(1+n2)n2sin(narctan(n))

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