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

• MHB
• Deanmark
In summary: The second equation is a polynomial in $w_0$, and so it has a unique solution that is also a complex number. Therefore, $u$ satisfies the conditions.
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.

Deanmark said:
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?

castor28 said:
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 ?

Deanmark said:
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 $|u|>1/\delta$.

## 1. What is a complex function?

A complex function is a type of mathematical function that involves complex numbers, which are numbers that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Complex functions can take in a complex number as input and produce a complex number as output.

## 2. What does the function f(z) = e^1/z mean?

This function means that the value of f(z) is equal to the complex number e (Euler's number) raised to the power of 1 divided by z, where z is a complex number. In other words, f(z) is equal to e^(1/z).

## 3. How do you prove that f(z) = e^1/z is true for all complex numbers?

To prove that a complex function is true for all complex numbers, you can use mathematical techniques such as proof by induction or proof by contradiction. In the case of f(z) = e^1/z, you can use the definition of a complex function and properties of complex numbers to show that it holds true for all complex numbers.

## 4. What is the significance of the function f(z) = e^1/z?

The function f(z) = e^1/z has many applications in mathematics, physics, and engineering. It is often used in the study of complex analysis and is useful for solving differential equations. It also has connections to other important mathematical concepts such as the complex logarithm and the Riemann surface.

## 5. Can you graph the function f(z) = e^1/z?

No, it is not possible to graph f(z) = e^1/z in the traditional sense because it is a complex function. However, you can graph the real and imaginary parts of the function separately on a complex plane to get a better understanding of its behavior. The graph will show that the function has an essential singularity at z = 0, and its values increase rapidly as z approaches 0 from any direction.

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