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GATE 2014 EE (EE2) – Question 14

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy integral formula, Taylor series, Laurent series, Residue theorem · 1 mark · Multiple choice

All values of the multivalued complex power $i^i$, with $i=\sqrt{-1}$, are ______.

  1. purely imaginary
  2. real and nonnegative
  3. on the unit circle
  4. equal in real and imaginary parts

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Show answer and explanation

Correct answer: (B) real and nonnegative

Explanation

The multivalued logarithm is log i=i(π/2+2πk),k integer. Hence i^i=exp[i log i]=exp(−π/2−2πk). Every value is a strictly positive real number, giving **real and nonnegative,B**. These values do not in general lie on the unit circle.