Cauchy's Integral Theorem (Fundamental Theorem for Complex Integrals):

Cauchy's Integral Theorem states that if $f(z)$ is holomorphic (analytic) on and inside a simple closed contour $C$, then the integral of $f(z)$ along $C$ is zero. In other words, if $f(z)$ is analytic everywhere inside and on a simple closed curve $C$, then:

$\oint_C f(z) \, dz = 0$

Conditions for the theorem to hold:

1. The function $f(z)$ must be analytic (holomorphic) inside and on the contour $C$.
2. The contour $C$ must be closed and simple, meaning it doesn't intersect itself and encloses a well-defined region.

Justification for the function $f(z) = z^3 + i z^2 - 5z + 2i$ over the circle $|z| = 1$:

Let's check if Cauchy's Integral Theorem applies to the function $f(z) = z^3 + i z^2 - 5z + 2i$, where $C$ is the circle $|z| = 1$.

1. Holomorphy: The function $f(z) = z^3 + i z^2 - 5z + 2i$ is a polynomial in $z$, and polynomials are entire functions. An entire function is holomorphic everywhere in the complex plane, including inside and on the contour $C$.

   Since $f(z)$ is analytic on and inside the contour $|z| = 1$, it satisfies the conditions for Cauchy's Integral Theorem.

2. Conclusion based on the theorem: Since $f(z)$ is holomorphic inside and on the contour $C$, Cauchy's Integral Theorem applies. Therefore, the integral of $f(z)$ over the circle $|z| = 1$ must be zero:

$\oint_C f(z) \, dz = 0$

This result holds because $f(z)$ is analytic everywhere in the complex plane, satisfying all the conditions for the application of Cauchy's theorem.

Final Answer:

For the function $f(z) = z^3 + i z^2 - 5z + 2i$ and the contour $C$ being the circle $|z| = 1$, the integral of $f(z)$ around $C$ is zero based on Cauchy's Integral Theorem:

$\oint_C f(z) \, dz = 0$

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Would you like further details or clarification on any part of this explanation?

Here are 5 follow-up questions to deepen your understanding:

1. How do you determine if a function is holomorphic in general?
2. Can Cauchy's Integral Theorem apply if a function has a singularity inside the contour $C$?
3. What are some consequences of Cauchy's Integral Theorem in complex analysis?
4. How would you compute the integral if $f(z)$ had singularities inside the contour $C$?
5. What is the difference between Cauchy's Integral Theorem and Cauchy's Integral Formula?

Tip: When working with complex integrals, always check if the function is analytic within the region enclosed by the contour—this is crucial for applying Cauchy's theorems.