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| source: adobe stock |
If n is an integer, positive or negative and Θ is a real number
(COSΘ +iSINΘ)∧n = cosnθ+isinnθ
If n is a fraction, positive or negative and Θ is a real number cosnθ+isinnθ is one of the values of (COSΘ +iSINΘ)∧n
More explicitly,
( i ) If n is an integer, positive or negative and Θ is a real number, then (COSΘ +iSINΘ)∧n has only one value, namely cosnθ+isinnθ
( ii ) If n is a positive fraction of the form p / q in its lowest terms, p and q being positive integers
( q≠ 0 ) and Θ is a real number, then (COSΘ +iSINΘ)∧(p/q) has q values of which one value is cos(p/q)θ+isin(p/q)θ
(iii)If n is a negative fraction of the form (-p/q) in its lowest terms, p and q being positive integers ( q≠0) and Θ is a real number, then (COSΘ +iSINΘ)∧(-p/q) has q values of which one value is
cos(-p/q)θ+isin(-p/q)θ=cos(p/q)θ-isin(p/q)θ
(iv) If n is an irrational number and θis a real number, then (COSΘ +iSINΘ)∧n has an infinite number of values of which one value is cosnθ+isinnθ
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