1 mathematics Q (1)

ni is a complex no. which can be written in exp. form keiθ where k and θ are both real, then:

ni = keiθ

Taking natural log to both sides:

i ln n = ln k + iθ

Thus if n = 0, the expression ni is undefined

If n > 0:

i ln n = ln k + iθ

Comparing real and imag. parts:

ln k = 0, giving k = 1

θ = ln n

Hence ni = keiθ = ei ln n = cos (ln n) + i sin (ln n)

where the angle expressed as ln n is in radian

If n < 0:

i ln n = ln k + iθ

i ln [(-n) eiπ] = ln k + iθ

i [ln (-n) + iπ] = ln k + iθ

-π + i ln (-n) = ln k + iθ

Comparing real and imag. parts:

ln k = -π, giving k = e-π

θ = ln (-n)

Hence ni = keiθ = ei ln(-n) = (1/eπ) {cos [ln (-n)] + i sin [ln (-n)]}

where the angle expressed as ln (-n) is in radian