Variation of Argument

Let $[\arg f(z)]$ denote the change in argument of a function $f(z)$ around a closed loop $\gamma$. Also let $N$ denote the number of Roots of $f(z)$ in $\gamma$ and $P$ denote the number of Poles of $f(z)$ in $\gamma$. Then

$$[\arg f(z)]= {1\over 2\pi}(N-P).$$ (1)

To find $[\arg f(z)]$ in a given region $R$, break $R$ into paths and find $[\arg f(z)]$ for each path. On a circular Arc

$$z = Re^{i\theta},$$ (2)

let $f(z)$ be a Polynomial $P(z)$ of degree $n$. Then

$${[}\arg P(z)] = \left[{\arg\left({z^n {P(z)\over z^n}}\right)}\right]$$

$$= [\arg z^n] + \left[{\arg \left({P(z)\over z^n}\right)}\right].$$ (3)

Plugging in $z = Re^{i\theta}$ gives

$$[\arg P(z)]= [\arg Re^{i\theta n}] + \left[{\arg { P(Re^{i\theta})\over Re^{i\theta n}}}\right]$$ (4)

$$\lim_{R\to \infty} {P(Re^{i\theta})\over Re^{i\theta n}} = \hbox{[constant]},$$ (5)

so

$$\left[{ P(Re^{i\theta})\over Re^{i\theta n}}\right]= 0,$$ (6)

and

$$[\arg P(z)]= [\arg e^{i\theta n}] = n(\theta_2-\theta_1).$$ (7)

For a Real segment $z = x$,

$$[\arg f(x)]= \tan^{-1}\left[{0\over f(x)}\right]= 0.$$ (8)

For an Imaginary segment $z = iy$,

$$[\arg f(iy)]= \left\{{\tan^{-1} {\Im[P(iy)]\over \Re[P(iy)]}}\right\}_{\theta_1}^{\theta_2}.$$ (9)

Note that the Argument must change continuously, so ``jumps'' occur across inverse tangent asymptotes.