Example 5. Find the number of roots of the equation

$$
Q_{7}(z) \equiv z^{7}-2 z-5=0
$$

in the right half-plane.
Solution. We choose the contour $C$ as indicated in Example 4. Then $\Delta_{C_{R}} \operatorname{Arg} Q_{7}(z)=\Delta_{C_{R}} \operatorname{Arg}\left(z^{7}-2 z-5\right)=$

$$
\begin{aligned}
& =\Delta_{C_{R}} \operatorname{Arg}\left[z^{7}\left(1-\frac{2}{z^{6}}-\frac{5}{z^{7}}\right)\right]=7 \Delta_{C_{R}} \operatorname{Arg} z+\Delta_{C_{R}} \operatorname{Arg}\left(1-\frac{2}{z^{6}}-\frac{5}{z^{7}}\right)= \\
& =7 \pi+\Delta_{C_{R}} \operatorname{Arg}\left(1-\frac{2}{z^{6}}-\frac{5}{z^{7}}\right) \rightarrow 7 \pi \quad \text { as } \quad R \rightarrow \infty
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_f7d63c3a5e94c3f2f1bbg-113.jpg?height=393&width=439&top_left_y=1358&top_left_x=141)

Fig. 13

Let $z=$ it $(-R \leqslant t \leqslant R)$. Then

$$
Q_{7}(i t)=u(t)+i v(t)=-5+i\left(-t^{7}-2 t\right)
$$

from which

$$
\left\{\begin{array}{l}
u=-5 \\
v=-t\left(t^{6}+2\right)
\end{array}\right.
$$

Since $u \neq 0$, the application of the argument principle is valid ( $Q_{7}(z)$ has no zeros on the imaginary axis). This line is a straight line (Fig. 13). The vector $w=Q_{7}(z)$ makes a turn in the negative direction by $\pi$ radians. Therefore,

$$
\Delta_{\dot{c}_{R}} \operatorname{Arg} Q_{1}(z) \xrightarrow[R \rightarrow \infty]{\longrightarrow} 7 \pi-\pi=6 \pi
$$

and

$$
N=\frac{6 \pi}{2 \pi}=3
$$

## Problems for Independent Solution

Determine the number of roots in the right half-plane for the following equations:

435. $z^{4}+2 z^{3}+3 z^{2}+z+2=0$. 436. $z^{3}-2 z-5=0$.
436. $z^{3}-4 z^{2}+5=0$.
437. $2 z^{3}-z^{2}-7 z+5=0$.
438. $z^{5}+5 z^{4}-5=0$.

440: $z^{12}-z+1=0$.

Rouche's Theorem. Suppose functions $f(z)$ and $\varphi(z)$, analytic in the closed domain $\bar{D}$, bounded by the contour $C$, satisfy the inequality $|f(z)|>|\varphi(z)|$ at all points of this contour. Then their sum $F(z)=f(z)+\varphi(z)$ and the function $f(z)$ have the same number of zeros (counting multiplicities) in the domain $D$.