How to find the real part of this fraction?

Question

I don't understand how to get the real part of the following fraction.

$\quad\dfrac{1}{a + jw}$

Here, $j$ is the imaginary unit.

What's the process of retrieving the real part? Thanks.

Answer 1

Hint: If $a$ and $w$ are real numbers, then try multiplying the numerator and denominator of the fraction by the complex conjugate in the denominator, and simplify.

That is, compute and simplify: $$ \dfrac{1}{a + jw} \cdot \dfrac{a - jw}{a-jw} = \dfrac{a-jw}{a^2 + w^2}= \left(\dfrac{a}{a^2 + w^2}\right) + \left( -\dfrac{w}{a^2 + w^2}\right) \cdot j$$

Answer 2

You could start by multiplying both the numerator and the denominator by the complex conjugate of $a + jw$, i.e. $a - jw$.