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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!

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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!

Using Cauchy integral formula, find the value of ∫_C (e^z / (z - π)) dz where C is the circle |z - 1| = 4.

Learn to evaluate the contour integral ∫_C (e^z / (z - π)) dz where C is the circle |z - 1| = 4 using Cauchy's Integral Formula. This solution explains key concepts in complex analysis, including singularities and their evaluation.

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