This is from the page 19 of the book called An Introduction to Complex Function Theory by Bruce P. P

Alaina Holt

Alaina Holt

Answered question

2022-05-17

This is from the page 19 of the book called An Introduction to Complex Function Theory by Bruce P. Palka.
Every rational function of z=x+iy is clearly a rational function of x and y, but not the other way around.
As a reminder, a rational function of z=x+iy is defined as f ( z ) = a 0 + a 1 z + + a n z n b 0 + b 1 z + b m z m . I understand the first part of the claim: if z is fixed, i.e. both x and y are fixed, then of course all combinations of as and bs are part of the set where as, bs, xs and ys can vary. But I don't see why this wouldn't hold the other way around. Specifically, why cannot we choose the coefficients a 0 , , a n and b 0 , , b m in such a way that any change in x or y is mitigated?

Answer & Explanation

overnachtt9xyx

overnachtt9xyx

Beginner2022-05-18Added 14 answers

Take for example the function f ( z ) = x, the real part of z. Then f is a rational function of x but not of z since
f ( z ) = z + z ¯ 2 .
On the other hand, we may say that every rational function of x and y is a rational function of z and z ¯ and the other way around because
{ z = x + i y z ¯ = x i y { x = z + z ¯ 2 y = z z ¯ 2 i

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