Talk:Cohn's theorem
Latest comment: 1 year ago by X-Fi6 in topic Missing complex conjugate sign in the conjugate reciprocal formula
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Missing complex conjugate sign in the conjugate reciprocal formula
editIf 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 + … + anz−n
- ⇒ f(1/z) = a0 + … + anz−n
- ⇒ f(1/z) = a0 + … + anz−n
- ⇒ 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.)