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calculste-0-1-e-zx-x-dx-with-z-from-C-and-Re-z-gt-0-

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Question Number 90291 by mathmax by abdo last updated on 22/Apr/20
calculste ∫_0 ^∞ ((1−e^(zx) )/x)dx with z from C and Re(z)>0
calculste∫0∞1−ezxxdxwithzfromCandRe(z)>0
Commented by mathmax by abdo last updated on 22/Apr/20
sorry Re(z)<0 .
sorryRe(z)<0.
Commented by mathmax by abdo last updated on 23/Apr/20
let f(z) =∫_0 ^∞ ((1−e^(zx) )/x)dx ⇒f^′ (z) =∫_0 ^∞ (∂/∂z)(((1−e^(zx) )/x))dx =−∫_0 ^∞ e^(zx) dx =−[(1/z)e^(zx) ]_0 ^(+∞) =−(1/z)(−1)=(1/z) (Rez<0) ⇒ f(z) =lnz +K so if z =α+iβ (α<0) ⇒z =(√(α^2 +β^2 ))e^(iarctan((β/α))) ⇒ ln(z) =(1/2)ln(α^2 +β^2 ) +iarctan((β/α)) ⇒ f(z)=(1/2)ln(α^2 +β^2 )+iarctan((β/α))+K rest to find K
letf(z)=∫0∞1−ezxxdx⇒f′(z)=∫0∞∂∂z(1−ezxx)dx=−∫0∞ezxdx=−[1zezx]0+∞=−1z(−1)=1z(Rez<0)⇒f(z)=lnz+Ksoifz=α+iβ(α<0)⇒z=α2+β2eiarctan(βα)⇒ln(z)=12ln(α2+β2)+iarctan(βα)⇒f(z)=12ln(α2+β2)+iarctan(βα)+KresttofindK

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