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 :)

  • 1
    $\begingroup$ Your second interpretation. $\endgroup$ – kimchi lover Feb 7 at 13:01

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 $\mathbb{Q}[x].$

  • $\begingroup$ Yes, the use of the word "operation" is what suggest. $\endgroup$ – DrinkingDonuts Feb 7 at 13:20
  • $\begingroup$ 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? :) $\endgroup$ – AfterMath Feb 7 at 13:21

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