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let-give-0-pi-prove-that-0-1-dt-e-i-t-n-1-e-in-n-

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Question Number 28611 by abdo imad last updated on 27/Jan/18
let give θ∈]0,π[ prove that ∫_0 ^1 (dt/(e^(−iθ) −t))= Σ_(n=1) ^(+∞) (e^(inθ) /n) .
letgiveθ∈]0,π[provethat∫01dte−iθ−t=∑n=1+∞einθn.
Commented by abdo imad last updated on 28/Jan/18
let put I= ∫_0 ^1 (dt/(e^(−iθ) −t)) ⇒ I= ∫_0 ^1 (e^(iθ) /(1−e^(iθ) t)) but we have ∣e^(iθ) t∣≤1 I= ∫_0 ^1 e^(iθ) ( Σ_(n=0) ^(+∞) e^(inθ) t^n )dt = Σ_(n=0) ^(+∞) e^(i(n+1)θ) ∫_0 ^1 t^n dt = Σ_(n=0) ^(+∞) (e^(i(n+1)θ) /(n+1))= Σ_(n=1) ^(+∞) (e^(inθ) /n) .
letputI=∫01dte−iθ−t⇒I=∫01eiθ1−eiθtbutwehave∣eiθt∣⩽1I=∫01eiθ(∑n=0+∞einθtn)dt=∑n=0+∞ei(n+1)θ∫01tndt=∑n=0+∞ei(n+1)θn+1=∑n=1+∞einθn.

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