Find the image of the infinite stripe 0 ≤ y ≤ π under the transformation w = ez.

To find the image of the infinite stripe \(0 \leq y \leq \pi\) under the transformation \( w = e^z \), let’s proceed with a step-by-step analysis:

1. **Rewrite \( z \) in terms of real and imaginary parts**: Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Apply the transformation \( w = e^z \)**:
   \[
   w = e^{x + iy} = e^x e^{iy}.
   \]
   Using Euler's formula \( e^{iy} = \cos(y) + i\sin(y) \), we get
   \[
   w = e^x (\cos(y) + i\sin(y)).
   \]

3. **Interpret the modulus and argument of \( w \)**:
   - The modulus \( |w| \) of \( w = e^x e^{iy} \) is \( |w| = e^x \).
   - The argument \( \arg(w) \) of \( w = e^x e^{iy} \) is \( y \).

4. **Determine the range for \( |w| \) and \( \arg(w) \)**:
   - Since \( x \) can take any real value, \( e^x \) spans all positive real numbers, meaning \( |w| > 0 \).
   - Since \( y \) is restricted by \( 0 \leq y \leq \pi \), the argument \( \arg(w) \) is constrained by \( 0 \leq \arg(w) \leq \pi \).

5. **Conclusion**: The image of the stripe \( 0 \leq y \leq \pi \) under the mapping \( w = e^z \) is the upper half of the complex plane, where
   \[
   0 < |w| < \infty \quad \text{and} \quad 0 \leq \arg(w) \leq \pi.
   \]
   This is equivalent to the set of all complex numbers \( w \) with positive real parts, forming the entire upper half of the complex plane (excluding the negative real axis).

Would you like more details or have any questions?

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1. How would the transformation change if \( w = e^{iz} \) instead?
2. What is the image of a different stripe, such as \( -\pi \leq y \leq \pi \), under \( w = e^z \)?
3. How does this transformation relate to mapping the unit disk in the complex plane?
4. Can we generalize the mapping \( w = e^z \) to other functions, like \( w = e^{az} \) for \( a \in \mathbb{C} \)?
5. What would happen if we considered \( z = u + iv \) and transformed the horizontal strip \( u \in [0,1] \) under \( w = e^z \)?

**Tip:** Transformations involving \( e^z \) often map regions of the complex plane onto sectors or disks, taking advantage of exponential growth and periodicity in the imaginary part.

Find the image of the infinite stripe 0 ≤ y ≤ π under the transformation w = e^z.

Explore the transformation of the infinite stripe 0 ≤ y ≤ π under w = e^z, detailing complex analysis principles and the use of Euler's formula. Learn how this maps to the upper half of the complex plane, focusing on modulus and argument behavior.

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