# What is meant by “rational operation”?

So I am attending a course in complex analysis and came across the following statement pretty early in the book:

Let $$R(a,b,c,\dots)$$ stand for any rational operation applied to the complex numbers $$a,b,c,\dots$$

Then $$\overline{R(a,b,c,\dots)}=R(\overline{a},\overline{b},\overline{c},\dots)$$

Where the bar denotes complex conjugation. [Lars Ahlfors, Complex analysis, 3rd. edition]

What exactly is meant by rational operation?

My first thought was that maby this meant that: $$R(a,b,c,\dots)=\frac{P(a,b,c,\dots)}{Q(a,b,c,\dots)}$$ Where $$P$$ and $$Q$$ are polynomials of (finite) degree in the varibales $$a,b,c,\dots$$

If my first interpretation is correct, then surely $$P$$ and $$Q$$ must have real coefficients, right?

My second thought was that maby this simply means a sequence of addition, subtraction, multiplication and division applied to $$a,b,c,\dots$$, with the assumption that no zero-divison takes place.

If not, what is the correct interpretation / meaning of the statement?

Thanx, R :)

• Your second interpretation. – kimchi lover Feb 7 at 13:01

That will give, simplifying the operations, a rational function, that is an element of the field of fractions $$\mathbb{Q}(x)$$ of the polynomial ring $$\mathbb{Q}[x].$$
• Yeah, It sounds the most resonalble, doesn't it? By the way, If I define $$R_n(a,b,c,\dots)$$ as the operation applied to only the first $n$ complex numbers in $a,b,c, \dots$, then I can easily show that $$\overline{R_n(a,b,c,\dots)}= R_n(\overline{a},\overline{b},\overline{c},\dots)$$ for all $n \in \mathbb{N}$. Now, given that the sequence of operations converge to some complex number, then I would be able to show that $$\overline{ R(a,b,c,\dots) }= \overline { \lim_{n \to \infty} R_n(a,b,\dots) } =\lim_{n \to \infty} R_n(\overline{a},\dots) =R(\overline{a},\dots)$$ , maby? :) – AfterMath Feb 7 at 13:21