Complex Variables and Partial Differential Equations (3130005)

Question-2b

BE | Semester-3 Winter-2019 | 26-11-2019

Q2) (b)

Define $\log z$ , prove that $i^{i} = e^{- (4 n + 1) \frac{π}{2}}$ .

Let,

z = r e^{iθ}

( Exponential Form )

\Rightarrow \log z = \log r + \log e^{iθ}

\Rightarrow \log z = \log r + i θ \log e

\Rightarrow \log z = \log r + i θ - - - - - (1)

\Rightarrow \log z = \log r; r = \sqrt{x^{2} + y^{2}} & θ = argument of z

Let,

z = {(i)}^{i}

\Rightarrow \log z = \log {(i)}^{i}

\Rightarrow \log z = i \times \log i

\Rightarrow \log z = i \times i (\frac{π}{2} + 2 kπ) [\log (i) = i (\frac{π}{2} + 2 kπ) as z = i & [1]]

\Rightarrow \log z = i^{2} (\frac{π}{2} + 2 kπ)

\Rightarrow \log z = - (\frac{π}{2} + 2 kπ)

\Rightarrow z = e^{- (\frac{π}{2} + 2 kπ)}

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