If x-1- then the value of xx is

X^x=i^i Let z=i^i Taking log on both sides nbsp; log z= i log i =i [ 12log(0^2+1^2)+i tan^ 110 ] =i [ iπ2 ]= π2 Z=e^ π2 Method II: x^x=i^i= ( e^iπ/2 )^i=e^ ...

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Engineering Mathematics

Complex Function ( Limit, Analytic Function)

If x=√-1, the...

Question

If $x = \sqrt{- 1}$ , then the value of $x^{x}$ is

e^{- π / 2}

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e^{- π / 2}

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Solution

The correct option is A $e^{- π / 2}$
$x^{x} = i^{i}$
Let $z = i^{i}$
Taking log on both sides
$l o g z = i l o g i$
$= i [\frac{1}{2} l o g (0^{2} + 1^{2}) + i t a n^{- 1} \frac{1}{0}]$
$= i [i \frac{π}{2}]$ = $\frac{- π}{2}$
$Z = e^{- \frac{π}{2}}$
Method - II:
$x^{x} = i^{i} = {(e^{\frac{i π}{2}})}^{i} = e^{- \frac{π}{2}}$

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If x=√−1, then the value of xx is

The correct option is A e−π/2xx=ii Let z=ii Taking log on both sides logz=ilogi =i[12log(02+12)+itan−110] =i[iπ2]=−π2 Z=e−π2 Method - II: xx=ii=(eiπ2)i=e−π2

If $x = \sqrt{- 1}$ , then the value of $x^{x}$ is