(1) Let r > 0 and let
A = {w | w = exp 1/z

, where 0 < |z| < r}.
Determine the set A.

Given the problem, we need to determine the set $ A $ defined as:

\[
A = \left\{ w \ \middle|\ w = \exp\left(\frac{1}{z}\right), \text{ where } 0 < |z| < r \right\}.
\]

Here, $ z $ is a complex number, $ |z| $ denotes the modulus (or absolute value) of $ z $, and $ r > 0 $ is a given positive real number.

### Step 1: Analyzing the expression for $ w $
We are given:

\[
w = \exp\left(\frac{1}{z}\right)
\]

where $ \exp(x) $ denotes the exponential function $ e^x $. 

### Step 2: Understanding the range of $ w $
Given $ z $ is a complex number such that $ 0 < |z| < r $, the quantity $ \frac{1}{z} $ can take on all complex values with modulus greater than $ \frac{1}{r} $. In other words, $ \frac{1}{z} $ will map the annulus $ 0 < |z| < r $ to the entire complex plane, except for a disk of radius $ \frac{1}{r} $ centered at the origin.

So $ \frac{1}{z} $ takes values in the set:

\[
\left\{ \frac{1}{z} \ \middle|\  |z| < r \right\} = \{ \zeta \in \mathbb{C} \mid |\zeta| > \frac{1}{r} \}
\]

### Step 3: Applying the exponential function
The exponential function $ \exp(x) = e^x $ is entire and maps the complex plane (except for one point) to the punctured complex plane (i.e., the complex plane without the origin). However, it is not injective, since $ \exp(x + 2\pi i k) = \exp(x) $ for any integer $ k $.

Thus, $ w $ will range over all possible complex numbers except 0. This is because the exponential function never yields 0 for any finite input.

### Step 4: Conclusion
The set $ A $ is the set of all non-zero complex numbers:

\[
A = \mathbb{C} \setminus \{0\}
\]

This is because, as $ z $ varies such that $ 0 < |z| < r $, $ w = \exp\left(\frac{1}{z}\right) $ will cover all complex numbers except 0.

### Do you want more details or have any questions?

Here are 5 related questions:
1. What are the properties of the exponential function in the complex plane?
2. How does the exponential map relate to covering spaces in complex analysis?
3. Can you explain the significance of the region $ 0 < |z| < r $ in complex analysis?
4. What is the concept of a branch cut, and how does it apply to complex logarithms?
5. How does the image of a punctured disk under the exponential map relate to the entire complex plane?

**Tip:** When analyzing sets defined by complex functions, it's useful to consider the entire function's range and any special properties, like injectivity or periodicity.

Explore the set A in complex analysis defined by $A = \left\{ w \mid w = \exp\left(\frac{1}{z}\right), 0 < |z| < r \right\}$. Learn about the exponential function's role in mapping the annulus $0 < |z| < r$ to the entire complex plane (excluding 0).

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