Prove the Following is True About the Complex Function f(z) = e^1/z

[Deanmark]

Consider the function $f(z) = e^{1/z}$,
Show that for any complex number ${w}_{0} \ne 0$ and any δ > 0, there exists ${z}_{0} ∈ C$
such that $ 0 < |{z}_{0}| < δ$ and $f({z}_{0}) = {w}_{0}$

I really don't know where to begin on this.

 

[castor28]

Consider the function $f(z) = e^{1/z}$,
Show that for any complex number ${w}_{0} \ne 0$ and any δ > 0, there exists ${z}_{0} ∈ C$
such that $ 0 < |{z}_{0}| < δ$ and $f({z}_{0}) = {w}_{0}$

I really don't know where to begin on this.

Hi,

You could start by writing $z=1/u$; given $w_0$, you must now find a complex number $u$ with $|u|>1/\delta$ and $e^u=w_0$, which means that $|u|$ can be arbitrarily large.

You should try to solve separately for the absolute value and the argument of $w_0$. What is the locus of the points $u$ such that $e^u$ is constant in the complex plane? What is the locus of the points with $\arg e^u$ constant?

Can you prove that there exist arbitrarily large $u$ that satisfy both conditions?

 

[Deanmark]

Hi,

You could start by writing $z=1/u$; given $w_0$, you must now find a complex number $u$ with $|u|>1/\delta$ and $e^u=w_0$, which means that $|u|$ can be arbitrarily large.

You should try to solve separately for the absolute value and the argument of $w_0$. What is the locus of the points $u$ such that $e^u$ is constant in the complex plane? What is the locus of the points with $\arg e^u$ constant?

Can you prove that there exist arbitrarily large $u$ that satisfy both conditions?

Should we let the complex number u = $\frac{2}{\delta}$ + arg(${w}_{0})$i ?

 

[castor28]

Should we let the complex number u = $\frac{2}{\delta}$ + arg(${w}_{0})$i ?

Hi Deanmark,

You are almost there, but not quite. If $u=x+yi$, then $e^u=e^x\cdot e^{yi}$, and we have $|e^u| = |e^x|$ and $\arg e^u = y$ for any integer $n$.

To have $e^u=w_0$, you must have:

$$\begin{align*}
x &= \ln w_0\\
y &= \arg w_0 + 2n\pi
\end{align*}$$
​

where $n$ is any integer (because the argument is defined up to a multiple of $2\pi$, or, to put it otherwise, because $e^{2\pi i}=1$).

The first equation describes a vertical line. By taking $n$ large enough (for example, $n>\dfrac{1}{2\pi\delta})$, you can ensure that $|u|>1/\delta$.