To analyze the contour integral of \( f(z) = \tan\left(\frac{1}{4}z\right) \) around the unit circle \( C \), and determine whether Cauchy's Integral Theorem applies, we'll go through similar steps as before.

### 1. Definition and Context

Consider the contour integral around the unit circle \( C \) in the complex plane:
\[
\oint_C f(z) \, dz = \oint_C \tan\left(\frac{1}{4}z\right) \, dz
\]

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

### 2. Analyzing \( f(z) = \tan\left(\frac{1}{4}z\right) \)

The function \( f(z) = \tan\left(\frac{1}{4}z\right) \) is defined as the ratio of sine and cosine:
\[
f(z) = \frac{\sin\left(\frac{1}{4}z\right)}{\cos\left(\frac{1}{4}z\right)}
\]
The function \( \tan\left(\frac{1}{4}z\right) \) will have singularities (poles) where \( \cos\left(\frac{1}{4}z\right) = 0 \).

### 3. Locating the Singularities

The cosine function \( \cos\left(\frac{1}{4}z\right) \) equals zero at:
\[
\frac{1}{4}z = \frac{\pi}{2} + n\pi \quad \text{for integers } n
\]
This simplifies to:
\[
z = 2\pi \left(n + \frac{1}{2}\right)
\]
The first few singularities occur at:
\[
z = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots
\]
However, since the unit circle is defined by \( |z| = 1 \), we only consider the singularities inside or on this circle. Therefore, the closest singularity is at \( z = 2\pi \times \frac{1}{2} = \pi \), which lies outside the unit circle.

### 4. Cauchy's Integral Theorem

Cauchy's Integral Theorem applies if the function is analytic within the domain bounded by \( C \) and on \( C \) itself. In this case, since the singularities of \( \tan\left(\frac{1}{4}z\right) \) lie outside the unit circle, \( f(z) \) is analytic inside and on the unit circle.

### 5. Conclusion

Since \( \tan\left(\frac{1}{4}z\right) \) is analytic on and inside the unit circle, Cauchy's Integral Theorem applies. Therefore, the contour integral is zero:
\[
\oint_C \tan\left(\frac{1}{4}z\right) \, dz = 0
\]

### Questions for Further Exploration

1. What happens if a singularity lies exactly on the contour \( C \)?
2. How would the integral change if \( C \) enclosed a singularity of \( \tan\left(\frac{1}{4}z\right) \)?
3. Can we use the residue theorem to evaluate this integral if a singularity were inside \( C \)?
4. What are the locations of singularities for the function \( \tan\left(\frac{k}{z}\right) \) for different values of \( k \)?
5. How does the integral change if we change the contour from the unit circle to another closed curve?

**Tip:** When dealing with trigonometric functions in complex analysis, always consider the poles introduced by the zeros of the denominator function.

Explore the contour integral of f(z) = tan(1/4 z) around the unit circle in complex analysis. Understand the application of Cauchy's Integral Theorem, the locations of singularities, and the analytic properties of the function. Suitable for advanced undergraduate students and beyond.

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