Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Quiz 7 => Topic started by: Jiaqi Bi on December 17, 2020, 06:51:38 AM

Question: Using argument principle along line on the picture, calculate the number of zeroes of the following function in the left halfplane:
$z+a=e^z, (a>0)$
(The graph will be attached.)
(Plus: I think Professor Victor originally wants to give us $a>1$ rather than $a>0$?)
Solution:
$\begin{align}
h(iy)&=iy+ae^{iy}\\
\nonumber &=iy+acos(y)isin(y)\\
\nonumber &=(acos(y))+i(ysin(y))
\end{align}
$
If $a>0$, we cannot conclude anything for $\text{Re} h(iy)$, but if $a>1$, then $\text{Re} h(iy)$ is always positive because the range of $cos(y)$ is consistently from $1$ to $1$.
$\text{Im}h(iy)$ will increase when $y$ goes from $R$ to $R$.
$\begin{align}
h(\text{Re}^{it})=\text{Re}^{it}+a=e^{\text{Re}^{it}}
\end{align}
$
Where t is from $\frac{\pi}{2}$ to $\frac{3\pi}{2}$, and $z$ goes from $iR$ to $iR$. $h(z)$ in this circumstance has been travelled a counterclockwise circuit.
Therefore, the argument for $h(z)$ should be $2\pi$.
By The Argument Principle, $\frac{1}{2\pi}\cdot 2\pi=1$.
Hence, $h(z)$ has a total of one zero in this plane.

Question: Using argument principle along line on the picture, calculate the number of zeroes of the following function in the left halfplane:
$z+a=e^z, (a>0)$
(Plus: I think Professor Victor originally wants to give us $a>1$ rather than $a>0$?)
(The graph will be attached.)
Solution:
$\begin{align}
h(iy)&=iy+ae^{iy}\\
&=iy+acos(y)isin(y)\\
&=(acos(y))+i(ysin(y))
\end{align}
$
If $a>0$, we cannot conclude anything for $\text{Re} h(iy)$, but if $a>1$, then $\text{Re} h(iy)$ is always positive because the range of $cos(y)$ is consistently from $1$ to $1$.
$\text{Im}h(iy)$ will increase when $y$ goes from $R$ to $R$.
$\begin{align}
h(\text{Re}^{it})=\text{Re}^{it}+a=e^{\text{Re}^{it}}
\end{align}
$
Where t is from $\frac{\pi}{2}$ to $\frac{3\pi}{2}$, and $z$ goes from $iR$ to $iR$. $h(z)$ in this circumstance has been travelled a counterclockwise circuit.
Therefore, the argument for $h(z)$ should be $2\pi$.
By The Argument Principle, $\frac{1}{2\pi}\cdot 2\pi=1$.
Hence, $h(z)$ has a total of one zero in this plane.