For which z is the following true: Log(iz^2)=frac(i\pi)(2)+2Log(z)

Makaila Simon

Makaila Simon

Answered question

2022-09-17

For which z is the following true: L o g ( i z 2 ) = i π 2 + 2 L o g ( z ). We raised both sides by e and concluded this equation holds for all z 0, but are not sure if we have all the work for it. We are mostly concerned with how the Arg part comes into play here and if that will limit what z can be.

Answer & Explanation

Rachael Conner

Rachael Conner

Beginner2022-09-18Added 8 answers

Substitute z = r e i ϕ with π < ϕ π
Then we get:
Log ( i r 2 e 2 i ϕ ) = i π 2 + 2 Log ( r e i ϕ )
2 log r + Log ( e i ( π / 2 + 2 ϕ ) ) = i π 2 + 2 log r + 2 i ϕ
Arg ( e i ( π / 2 + 2 ϕ ) ) = π 2 + 2 ϕ
π < π 2 + 2 ϕ π
3 π 4 < ϕ π 4
So the expression holds true for any z 0 with 3 π 4 < Arg  z π 4

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