On p. 70 in Griffith's "Introduction to Algebraic Curves", the author states in a proof of a Lemma regarding the construction of a polynomial as a Weierstrass Polynomial, the Newton Symmetric Polynomials, $\sigma_k(x)$ are holomorphic as a consequence of the residue theorem and equal to the result given in the image below. I'm not sure how this derivation of the result given in the text comes from the residue theorem. Can anyone walk me through this in some detail? Please note: The text does not state the Residue Theorem
page 69:
page 70:
The Residue Theorem (in fact the Argument Principle) just gives $$ \sigma_k(x) = \frac{1}{2\pi i} \int_{|y|=\epsilon}y^k \frac{f_y(x,y)}{f(x,y)}dy $$ Now we use that $f$ is holomorphic to apply Leibniz rule to show that $\sigma_k$ is holomorphic and its derivative is $$ \frac{d\sigma_k(x)}{dx} = \frac{1}{2\pi i} \int_{|y|=\epsilon}y^k \frac{\partial}{\partial x}\frac{f_y(x,y)}{f(x,y)}dy = \frac{1}{2\pi i} \int_{|y|=\epsilon}y^k \frac{f_{xy}(x,y)f(x,y) - f_x(x,y)f_y(x,y)}{f(x,y)^2}dy $$