To solve the given problem using Cauchy's Integral Formula, let's analyze the problem step by step.

The integral is:

$\int_C \frac{e^z}{z - \pi} \, dz$

where $C$ is the circle $|z - 1| = 4$.

Step 1: Identify the Function and Pole

The integrand is:

$f(z) = \frac{e^z}{z - \pi}$

Here, the function $f(z)$ has a singularity at $z = \pi$. Since $C$ is the circle of radius 4 centered at $z = 1$, the distance from the center of the circle to $z = \pi$ is:

$|z - \pi| = |\pi - 1| = 3.14 - 1 = 2.14$

Since $2.14 < 4$, the singularity $z = \pi$ lies inside the circle.

Step 2: Apply Cauchy's Integral Formula

Cauchy's Integral Formula states that for a function $f(z)$:

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

Here:

• $f(z) = e^z$
• $a = \pi$

So, $f(a) = e^{\pi}$.

Step 3: Substitute into the Formula

Using the formula:

$\int_C \frac{e^z}{z - \pi} \, dz = 2\pi i \cdot e^{\pi}$

Final Answer:

$\int_C \frac{e^z}{z - \pi} \, dz = 2\pi i e^{\pi}$

Let me know if you would like a more detailed explanation or further clarification!

Here are 5 related questions for better understanding:

1. How does Cauchy's Integral Formula simplify computations of contour integrals?
2. What happens if the singularity lies outside the contour $C$?
3. Can this integral be solved without using Cauchy's Integral Formula? How?
4. How would the result change if $C$ was a different contour not enclosing $z = \pi$?
5. What are the geometric implications of the circle $|z - 1| = 4$ in the complex plane?

Tip: Always check whether the singularity is inside or outside the contour before applying Cauchy's Integral Formula!