Missing complex conjugate sign in the conjugate reciprocal formula

edit

If f(z) = a0 + … + anzn is a polynomial, then its conjugate reciprocal is given by reversing and conjugating the coefficients, i.e., f*(z) = an + … + a0zn.

The formula for conjugate reciprocal (when z≠0) should be f*(z) = znf(1/z), i.e., there should be a complex conjugate sign on both the input and output of f.

Steps:

f(z) = a0 + … + anzn
f(1/z) = a0 + … + anzn
f(1/z) = a0 + … + anzn
f(1/z) = a0 + … + anzn
znf(1/z) = a0zn + … + an =: f*(z).

The article was missing a complex conjugate on the input of f, so I added it in. This matches the formula used in the Lehmer–Schur algorithm article.

Arthur Cohn made the same mistake all throughout his 1921 PhD dissertation 😄. [1] (See e.g. equation (1) on page 112.)

[1] https://gdz.sub.uni-goettingen.de/id/PPN314393552

X-Fi6 (talk) 01:45, 4 December 2022 (UTC)Reply