Question

Suppose z is a complex number and |z|= 4 ,
arg(z)=π2
arg(z)=π2
then z = ...

Answer 1

Answer on Question #58294 – Math – Complex Analysis

Question

Suppose $z$ is a complex number and $|z| = 4$,

$$\arg(z) = \pi/2$$

then $z =$ ...

Solution

The exponential form of the complex number $z$ is given by

$$z = |z| e^{i \arg(z)} = 4 e^{i \frac{\pi}{2}} = 4 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = 4i,$$

because

$$e^{i\theta} = \cos(\theta) + i \sin(\theta)$$

according to Euler's formula.

Answer: $z = 4i$.

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