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sing the relation z=x+iy = rcosθ + irsinθ=re^iθ . Derive z^n = r^n(cosθ + isinθ)^n = r^n(cosnθ + isinnθ). Please show full and complete procedure

sing the relation z=x+iy = rcosθ + irsinθ=re^iθ . Derive z^n = r^n(cosθ + isinθ)^n = r^n(cosnθ + isinnθ). Please show full and complete procedure

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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sing the relation z=x+iy = rcosθ + irsinθ=re^iθ . Derive z^n = r^n(cosθ + isinθ)^n = r^n(cosnθ + isinnθ). Please show full and complete procedure 

Expert Solution
Step 1: Given the information

The relation z equals x plus i y equals r cos theta plus i r sin theta equals r e to the power of i theta end exponent.

The aim is to derive the expression z to the power of n equals r to the power of n open parentheses cos theta plus i sin theta close parentheses to the power of n equals r to the power of n open parentheses cos open parentheses n theta close parentheses plus i sin open parentheses n theta close parentheses close parentheses.

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