To compute the integral

$\oint_{C[0,2]} \frac{\sin(4z)}{4z - \pi^2} \, dz$

using the Cauchy integral formula, we need to follow these steps:

1. Identify the form of the Cauchy integral formula to be used:
   The Cauchy integral formula states that for a function $f(z)$ analytic inside and on a simple closed contour $C$, and for $a$ inside $C$,

   $f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - a} \, dz.$

   We can rearrange this formula to solve for the integral:

   $\oint_C \frac{f(z)}{z - a} \, dz = 2\pi i f(a).$

2. Identify $f(z)$ and $a$:
   By comparing the given integral with the Cauchy integral formula, we identify:

   • $f(z) = \sin(4z)$
   • $a = \frac{\pi^2}{4}$

   The integral then becomes:

   $\oint_{C[0,2]} \frac{\sin(4z)}{z - \frac{\pi^2}{4}} \, dz.$

3. Check if $a = \frac{\pi^2}{4}$ is inside the contour $C[0,2]$:
   The contour $C[0,2]$ is a circle centered at 0 with radius 2. The value $\frac{\pi^2}{4} \approx 2.467$ which lies outside this contour (since $\pi^2 \approx 9.87$ and $\frac{\pi^2}{4} > 2$).

Since $a = \frac{\pi^2}{4}$ is outside the contour $C[0,2]$, the integrand has no singularity inside the contour. According to the Cauchy-Goursat theorem, the integral of an analytic function over a closed contour that encloses no singularities is zero.

Thus,

$\oint_{C[0,2]} \frac{\sin(4z)}{4z - \pi^2} \, dz = 0.$

Summary

The integral evaluates to zero because the singularity $\frac{\pi^2}{4}$ lies outside the contour $C[0,2]$.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How would the result change if the singularity were inside the contour?
2. Can you provide an example where the singularity is inside the contour?
3. What is the geometric interpretation of the Cauchy integral formula?
4. How does the Cauchy integral formula relate to residues?
5. What happens if the function $f(z)$ has multiple singularities?

Tip: When applying the Cauchy integral formula, always ensure to verify whether the singularity lies inside or outside the contour to correctly determine the value of the integral.