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.