To solve the integral $\int_0^{2\pi} \cos^3\left(3 + \frac{1}{z-3}\right) \, dz$, where $z$ is a complex variable, we can approach it using complex analysis.

Step 1: Analyze the function

The integrand $\cos^3\left(3 + \frac{1}{z-3}\right)$ is a complex function. The key point to notice is the singularity at $z = 3$, where the denominator $z - 3$ becomes zero. This point is a pole of the integrand.

Step 2: Consider the contour integral

To evaluate the integral using contour integration, we express it as a contour integral in the complex plane:

$\oint_{\Gamma} \cos^3\left(3 + \frac{1}{z-3}\right) \, dz$

where $\Gamma$ is the contour around the unit circle $|z| = 1$.

Step 3: Apply Residue Theorem

Since the integral is over a closed contour and the function $\cos^3\left(3 + \frac{1}{z-3}\right)$ has a singularity at $z = 3$, which is outside the contour $|z| = 1$, the integral of this function over the closed contour is zero by the Residue Theorem. However, the integral given is from 0 to $2\pi$, not a contour integral.

Step 4: Conclusion

Given the information, it appears that the integral would be zero because it relates to a periodic function over a complete period from $0$ to $2\pi$, especially since the singularity at $z = 3$ lies outside the path of integration in the complex plane.

Therefore:

$\int_0^{2\pi} \cos^3\left(3 + \frac{1}{z-3}\right) \, dz = 0$

This result leverages the symmetry and periodic nature of the cosine function over the interval from $0$ to $2\pi$.

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Follow-up Questions:

1. What is the Residue Theorem and how is it applied in complex integration?
2. Why does the singularity at $z = 3$ not affect the integral from $0$ to $2\pi$?
3. How do periodic functions influence integrals over a full period?
4. What happens if the singularity lies within the contour of integration?
5. Can the method used here be applied to other trigonometric integrals in the complex plane?

Tip: When dealing with complex integrals, identifying singularities and understanding their positions relative to the contour is crucial for applying the Residue Theorem effectively.