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Question: 5]c) i) Considering only principal values separate into real and imaginary parts $i^{log(i+1)}$.
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Subject:- Applied Mathematics

Marks:- 3

Mumbai Unversity>FE>Sem1>Applied Maths1

m1(81) • 9 views
$let Z = i^log⁡(i+1) ∴ logZ = log(1+i).logi$ $But log(i+1) = log√2 + itan^{-1}⁡1 = log√2 + iπ/4 and logi = i.( π)/2$ $∴ logZ = (log√2 + iπ/4 ). i.( π)/2 = \frac{[1/2 log2+i π/4 π)}{2} = (-π)^2/8+i π/4 log2 = e^{-π^2/8+iθ} = e^{-π^2/8.iθ} where θ= π/4 log2 = e^{-π^2/8}[cosθ+isinθisinθ$ $∴ Real part of Z = e^{-π^2/8}cosθ= e^{-π^2/8}cos(π/4 log2)∴nary part of Z = e^{-π^2/8}sin(π/4 log2)$