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 :)
Since Ahlfors wrote "Rational operation applied..." I think that he means a sequence of sums, subtractions, products and quotients to the given complex numbers.
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 $\mathbb{Q}[x].$