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Question Number 189786 by sonukgindia last updated on 21/Mar/23

Answered by a.lgnaoui last updated on 22/Mar/23

A•z^z =(e^(iθ) )^e^(iθ)  =(cos θ+isin θ)^z   =e^(zlnz) =e^(e^(iθ) ln(e^(iθ)  )) =e^(ln[(e^(iθ) )^((cos θ+isin θ)) ])    =e^([ln(e^(iθcos θ) ×e^(−θsin θ) ]) =e^([ln(e^(iθcos θ) ×ln(e^(−θsin θ)) ])   =((ln(e^(iθcos θ) ))/(ln(e^(θsin θ) )))=((cos (θcos θ)+isin (θcos θ))/(θsin θ))  =((cos (θcos θ))/(θsin θ))+i((sin( θcos θ))/(θsin θ))  ⇒  z^z =e^([((cos (θcos θ)/(θsin θ))) ×e^(i((sin (θcos θ)/(θsin θ)))   =e^((cos (θcos θ))/(θsin θ)) [cos (((sin (θcos θ))/(θsin θ)))+sin (((sin( θcos θ)/(θsin θ)))]  ⇒    R(z^z )=e^((cos (θcos θ))/(θsin θ)) ×cos^((((sin (θcos θ))/(θsin θ))))

Azz=(eiθ)eiθ=(cosθ+isinθ)z=ezlnz=eeiθln(eiθ)=eln[(eiθ)(cosθ+isinθ)]=e[ln(eiθcosθ×eθsinθ]=e[ln(eiθcosθ×ln(eθsinθ)]=ln(eiθcosθ)ln(eθsinθ)=cos(θcosθ)+isin(θcosθ)θsinθ=cos(θcosθ)θsinθ+isin(θcosθ)θsinθzz=e[cos(θcosθθsinθ×eisin(θcosθθsinθ=ecos(θcosθ)θsinθ[cos(sin(θcosθ)θsinθ)+sin(sin(θcosθθsinθ)]R(zz)=ecos(θcosθ)θsinθ×cos(sin(θcosθ)θsinθ)

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