The Principle of the Argument

The Principle of the Argument

Theorem 1 (The Principle of the Argument): Let $A \subseteq \mathbb{C}$ be open, and let $f : A \to \mathbb{C}$ be analytic on $A$ except at finitely many poles in $A$. If $\gamma$ is any positively-oriented, simple, closed, piecewise smooth curve that is homotopic to a point in $A$ then $\displaystyle{\int_{\gamma} \frac{f'(z)}{f(z)} \: dz = 2\pi i ( Z_f^{\gamma} - P_f^{\gamma})}$ where $Z_f^{\gamma}$ is the number of roots of $f$ inside of $\gamma$ (counting multiplicities) and $P_f^{\gamma}$ is the number of poles of $f$ inside $\gamma$ (counting multiplicities).

For example, consider the function $f(z) = z^n$ where $n \in \mathbb{N}$, and let $\gamma$ be the positively-oriented unit circle. Then $f'(z) = nz^{n-1}$ and notice that:

(1)
\begin{align} \quad \int_{\gamma} \frac{f'(z)}{f(z)} \: dz = \int_{\gamma} \frac{nz^{n-1}}{z^n} = n \int_{\gamma} \frac{1}{z} \: dz = n 2\pi i \end{align}

We now use the principle of the argument to deduce the same result. Note that $f$ has $n$ roots inside the unit disk $D(0, 1)$ and has no poles inside $D(0, 1)$. So $Z_f^{\gamma} = n$ and $P_f^{\gamma} = 0$. Hence:

(2)
\begin{align} \quad \int_{\gamma} \frac{f'(z)}{f(z)} \: dz = 2\pi i (n - 0) = n 2\pi i \end{align}
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