Finding the conjugate of a complex number

complex numbers

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))

Best Answer

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.

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