Integrate counterclockwise around the unit circle.
Indicate whether Cauchy’s integral theorem applies. Show
the details.f(z)= exp(-z²)

To solve the integral of the function \( f(z) = \exp(-z^2) \) counterclockwise around the unit circle, we need to consider both the integral and whether Cauchy's Integral Theorem applies.

### 1. Definition and Context

Let's consider the contour integral around the unit circle \( C \) in the complex plane:
\[
\oint_C f(z) \, dz = \oint_C \exp(-z^2) \, dz
\]

Here, \( C \) is the unit circle centered at the origin, described by \( |z| = 1 \).

### 2. Cauchy's Integral Theorem

Cauchy's Integral Theorem states that if a function \( f(z) \) is analytic (holomorphic) throughout a simply connected domain \( D \) that contains the contour \( C \) and its interior, then the integral of \( f(z) \) around \( C \) is zero:
\[
\oint_C f(z) \, dz = 0
\]
This theorem only applies if \( f(z) \) is analytic in the domain containing \( C \) and the interior of \( C \).

### 3. Analyticity of \( f(z) = \exp(-z^2) \)

The function \( f(z) = \exp(-z^2) \) is an entire function, meaning it is analytic everywhere in the complex plane, including within and on the unit circle \( |z| = 1 \).

### 4. Applying Cauchy's Integral Theorem

Since \( f(z) \) is analytic everywhere, particularly within the domain containing the unit circle \( C \) and its interior, Cauchy's Integral Theorem directly applies. Therefore, the contour integral is zero:
\[
\oint_C \exp(-z^2) \, dz = 0
\]

### 5. Conclusion

The integral of \( \exp(-z^2) \) around the unit circle is zero by Cauchy's Integral Theorem, because the function is analytic everywhere in the complex plane.

### Questions for Further Exploration

1. What is the general form of Cauchy’s Integral Theorem?
2. How would the integral change if the function had a singularity inside the unit circle?
3. Can the result be extended to other contours besides the unit circle?
4. What happens if we integrate around a contour that encloses a singularity?
5. How does the result relate to the residue theorem?

**Tip:** Always check whether a function is analytic in the domain before applying Cauchy's Integral Theorem.

Learn how to integrate counterclockwise around the unit circle using Cauchy’s Integral Theorem with the function f(z) = exp(-z²). Understand the conditions under which the theorem applies and how analyticity affects the result. Suitable for advanced undergraduate students studying complex analysis.

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