Finding the conjugate of a complex number

Question

We know that the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. 

So, given a certain complex number, it is possibile to find its conjugate by writing it as:

Z = Re {Z} + j Im {Z}

and by considering:

Z* = Re {Z} - j Im {Z}

But in many applications (ex: signal theory etc) I saw people apply this rule: you have to replace "j" with "-j". Of course in case Z is written as shown before, it works. But in general?

For instance:

Z = (exp(4j)+sqrt(17j))/(exp(6j))

Answer 1

There are functions such that $(f(z))^*\neq f(z^*)$, for example, $$ f(z) = \mathrm{Re}z + \mathrm{Im}z.$$ However, if $f$ is analytic then the trick always works, i.e., $(f(z))^* = f(z^*)$. In your specific example the outcome will depend on how you choose to interpret the square root.

Answer 2

If you have an expression of the form $$ A + Bj, $$ where $A$ and $B$ are both real, then its conjugate is $$ A + B(-j) = A - Bj. $$

If your expression has any other form -- involving quotients, for instance, then the rule above does not apply, and may lead to an incorrect answer, as your example shows. (It may also lead to a correct answer, but it's not guaranteed to do so, and guarantees are what you need when doing mathematics.)

It's important, when you learn a trick like "just negate the j", that you also learn the context and assumptions of the trick. That's why theorems in mathematics all have hypotheses --- the things you have to check before applying the conclusion.