Complex integration of ∫_0^2π▒cos^3⁡(3+1/(z-3))dz

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.

Explore the complex integration of cos^3(3 + 1/(z-3)) from 0 to 2π using contour integration and the Residue Theorem. Understand how the singularity at z = 3 affects the integral and why the integral evaluates to zero over the full period. Suitable for advanced undergraduate students and those interested in complex analysis.

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