If z -1- i - i- then the correct options is-are -A- Re z - e -4cos log 2B- Imz-e-4sin1-2log 2Im z - e -4sin log 2D- Rez-e-4cos1-2log 2

Let z = (1 i)^ i, taking log on both sides log z = ilog_e (1 i) = ilog_e( √(2) ( cosπ4 isinπ4 )) = i log_e (√(2)e^ iπ4) = i [ 12log_e 2 +log_ee^ i π4 ] ...

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Byju's Answer

Standard XII

Mathematics

Logarithm of a Complex Number

If z =1- i - ...

Question

If $z = (1 - i)^{- i}$ , then the correct option(s) is/are

R e (z) = e^{- \frac{π}{4}} cos (log 2)

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I m (z) = - e^{- \frac{π}{4}} sin (\frac{1}{2} log 2)

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I m (z) = - e^{- \frac{π}{4}} sin (log 2)

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R e (z) = e^{- \frac{π}{4}} cos (\frac{1}{2} log 2)

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Solution

The correct options are
B $I m (z) = - e^{- \frac{π}{4}} sin (\frac{1}{2} log 2)$
D $R e (z) = e^{- \frac{π}{4}} cos (\frac{1}{2} log 2)$
Let $z = (1 - i)^{- i}$ , taking log on both sides
$log z = - i {log}_{e} (1 - i)$
$= - i {log}_{e} (\sqrt{2} (cos \frac{π}{4} - i sin \frac{π}{4}))$
$= - i {log}_{e} (\sqrt{2} e^{- i \frac{π}{4}})$
$= - i ⎡ ⎣ \frac{1}{2} {log}_{e} 2 + {log}_{e} e^{- i \frac{π}{4}} ⎤ ⎦$
$= - i [\frac{1}{2} {log}_{e} 2 - \frac{i π}{4}]$
$= - \frac{i}{2} {log}_{e} 2 - \frac{π}{4}$
$\Rightarrow z = e^{- \frac{π}{4}} \cdot e^{\frac{- i (log 2)}{2}}$
$\Rightarrow R e (z) = e^{- \frac{π}{4}} cos (\frac{1}{2} log 2), I m (z) = - e^{- \frac{π}{4}} sin (\frac{1}{2} log 2)$

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Similar questions

If $z = i^{i}$ ,where i = $\sqrt{- 1}$ , then Re(z) is

If $z = (i)^{i}$ , where $i = \sqrt{- 1}$ , then Re(z) is:

Q. The real part of

(1 - i)^{- i}

is
[RPET 1999]

Q. If

z = ∣ ∣ ∣ ∣ ∣ \begin{matrix} e^{2 i A} & e^{- i C} & e^{- i B} e^{- i C} & e^{2 i B} & e^{- i A} e^{- i B} & e^{- i A} & e^{2 i C} \end{matrix} ∣ ∣ ∣ ∣ ∣

, where

A + B + C = π

and

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then

Q. Assertion :If

z = \sqrt{3 + 4 i} + \sqrt{- 3 + 4 i}

, then principal arg of z i.e. arg (z) are

\pm \frac{π}{4}, \pm \frac{3 π}{4}

where

\sqrt{- 1} = i

. Reason: If z = A + iB, then

\sqrt{z} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \sqrt{\frac{| z | + R e (z)}{2}} + i \sqrt{\frac{| z | - R e (z)}{2}} & , if B > 0 \sqrt{\frac{| z | + R e (z)}{2} - i \sqrt{\frac{| z | - R e (z)}{2}}} & , if B < 0 \end{matrix}

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If z=(1−i)−i, then the correct option(s) is/are

If $z = (1 - i)^{- i}$ , then the correct option(s) is/are