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.