How to find the domain of a complex rational function? E.g. f ( z ) =

freakygirl838w

freakygirl838w

Answered question

2022-06-22

How to find the domain of a complex rational function?
E.g.
f ( z ) = 3 z i z
I understand that domain means z C for which this is defined, but I don't know what to look for.
Perhaps:
Cannot be z=0 because of divizion of zero. But is that all?
Are the criteria for "defined" the same as for real rational functions?

Answer & Explanation

Zayden Andrade

Zayden Andrade

Beginner2022-06-23Added 22 answers

z is defined everywhere and so is 3z−i. The only option for "undefinedness" is the division by zero.
More generally, polynomials are defined everywhere and rational fractions everywhere but at the zeroes of the denominator.
General rules:
1. z ± w and zw are defined wherever z and w are defined;
2. z w is defined where z and w are defined and w 0;
3. e z is defined wherever z is defined;
4. log z (main branch) is defined wherever z is defined but z=0.
This is about all you need to know, all the rest follows.

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