(1) Let r > 0 and let
A = {w | w = exp 
1
z

, where 0 < |z| < r}.
Determine the set A.

Given the set $ A = \left\{ w \mid w = \exp\left(\frac{1}{z}\right), \text{ where } 0 < |z| < r \right\} $, we need to determine the set $ A $.

### Analysis:

1. **Expression for $ w $:**
   \[
   w = \exp\left(\frac{1}{z}\right)
   \]
   where $ z $ is a complex number and $ 0 < |z| < r $.

2. **Domain of $ z $:**
   The modulus $ |z| $ represents the distance of $ z $ from the origin in the complex plane. The condition $ 0 < |z| < r $ means $ z $ lies in an open disk centered at the origin with radius $ r $, excluding the origin itself (since $ |z| > 0 $).

3. **Behavior of $ w $ as $ z $ varies:**
   - As $ z $ approaches 0 from any direction, $ \frac{1}{z} $ becomes arbitrarily large in magnitude (moving towards infinity in the complex plane). Therefore, $ \exp\left(\frac{1}{z}\right) $ will take on large values.
   - If $ z $ is close to 0, $ \frac{1}{z} $ will have a large real part, causing $ w $ to grow rapidly.
   - As $ |z| $ increases but stays within $ (0, r) $, $ \frac{1}{z} $ decreases in magnitude, so $ w = \exp\left(\frac{1}{z}\right) $ becomes smaller.

4. **Range of $ w $:**
   - Since $ \exp(z) $ maps any complex number $ z $ to the entire complex plane except for 0, $ w = \exp\left(\frac{1}{z}\right) $ can theoretically take any value in the complex plane except 0, given that $ z $ never actually reaches 0 (hence $ \frac{1}{z} $ never reaches infinity).
   - As $ |z| \to r $, the value $ \frac{1}{z} $ approaches $ \frac{1}{r} $, causing $ w $ to approach $ \exp\left(\frac{1}{r}\right) $.

### Conclusion:
The set $ A $ is the entire complex plane excluding the origin. This is because the exponential function $ \exp(z) $ is surjective over the complex plane (except for zero), and as $ z $ varies within the disk $ 0 < |z| < r $, $ w $ can take any complex value except 0.

Thus, the set $ A $ is:
\[
A = \mathbb{C} \setminus \{0\}
\]

Would you like more details or have any questions about this explanation? Here are a few related questions you might consider:

1. What is the significance of the exponential function in complex analysis?
2. How does the behavior of the exponential function differ on the real line versus in the complex plane?
3. Can the set $ A $ ever include the origin? Why or why not?
4. How does the condition $ 0 < |z| < r $ influence the range of $ w $?
5. What happens if the domain for $ z $ includes the origin?

**Tip:** The exponential function in the complex plane is a powerful tool, especially when dealing with transformations and mappings. Understanding its behavior can greatly enhance your comprehension of complex analysis.

Explore the set $A = \left\{ w \mid w = \exp\left(\frac{1}{z}\right), \text{ where } 0 < |z| < r \right\}$ in complex analysis. Understand the behavior of the exponential function in the complex plane and its range based on the condition $0 < |z| < r$. Suitable for advanced mathematics enthusiasts and students.

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