https://r2.erweima.ai/mathbot/image/1725896243211focwvi5l.jpg

The image shows the integral:

\[
\int e^{iz} \, dz = \frac{e^{iz}}{i} + C = -i e^{iz} + C
\]

### Explanation:

1. **Integral of \( e^{iz} \):**
   - The integral of \( e^{iz} \) follows the basic rule of integrating exponential functions, but since \( i \) is a constant (imaginary unit), we need to account for the derivative of the exponent \( iz \).
   - The derivative of \( iz \) with respect to \( z \) is \( i \), so the integral introduces a division by \( i \).
   - Thus:
     \[
     \int e^{iz} \, dz = \frac{e^{iz}}{i} + C
     \]
   
2. **Alternative Form:**
   - We can multiply both the numerator and the denominator by \( -1 \), leading to:
     \[
     \frac{e^{iz}}{i} = -i e^{iz}
     \]
   - Hence, the integral can also be expressed as:
     \[
     \int e^{iz} \, dz = -i e^{iz} + C
     \]
   - Both forms are valid, and the \( C \) represents the constant of integration.

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Would you like a more detailed explanation of this process or further clarification?

Here are 5 related questions to explore:
1. What is the integral of other exponential functions like \( e^{ax} \)?
2. How does the constant \( i \) affect integration results?
3. Can you extend this concept to integrals involving real and complex parts of functions?
4. What is the relationship between \( e^{iz} \) and trigonometric functions using Euler's formula?
5. How does the integration of complex functions differ from real functions?

**Tip:** When integrating exponential functions involving imaginary numbers, always pay attention to constants like \( i \), as they often modify the result through division or multiplication.

What is the integral of e^(iz) with respect to z?

Learn how to compute the integral of e^(iz) with respect to z. This example covers integration techniques involving exponential functions and imaginary numbers. Ideal for college students studying calculus or complex variables.

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