Logarithm of compex number

Algebra Level 2

Which of the following is a possible value of ln ⁡ ( 1 + i ) \ln(1+i) ?

ln ⁡ 2 2 + i π 4 \frac{\ln2}{2}+\frac{i\pi}{4} ln ⁡ 2 2 − i π 4 \frac{\ln2}{2}-\frac{i\pi}{4} ln ⁡ 2 4 + i π 2 \frac{\ln2}{4}+\frac{i\pi}{2} ln ⁡ 3 2 + i π 4 \frac{\ln3}{2}+\frac{i\pi}{4}

재환 하 by 재환 하 , 21, South Korea

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1 solution

재환 하
Dec 27, 2017

As the Euler's formula , e i x = c o s x + i s i n x e^{ix}=cosx+isinx

e i π 4 = c o s π 4 + i s i n π 4 = 2 2 ( 1 + i ) e^\frac{i\pi}{4}=cos\frac{\pi}{4}+isin\frac{\pi}{4}=\frac{\sqrt2}{2}(1+i)

∴ l n ( 1 + i ) = i π 4 + l n 2 2 ∴ ln(1+i)=\frac{i\pi}{4}+\frac{ln2}{2}

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