To solve the given problem, we'll follow the steps outlined in the image, making sure not to use Cauchy's Formula or the Residue Theorem. Let's analyze and solve the problem step-by-step.

Problem:

Calculate $\int_{|z|=3} \frac{e^z}{z^2 - 1} dz$ in the positive direction of the circle (opposite to the clockwise direction), first parameterizing the circle and then using that $z = 3 \left( \cos \theta + i \sin \theta \right) = 3 e^{i \theta}$, where $0 \leq \theta \leq 2\pi$.

Solution Steps:

1. Parameterize the Circle: $z = 3 e^{i \theta}, \quad dz = 3i e^{i \theta} d\theta, \quad 0 \leq \theta \leq 2\pi.$

2. Substitute in the Integral: $\int_{|z|=3} \frac{e^z}{z^2 - 1} dz = \int_{0}^{2\pi} \frac{e^{3 e^{i \theta}}}{(3 e^{i \theta})^2 - 1} \cdot 3i e^{i \theta} d\theta.$ Simplify the denominator: $(3 e^{i \theta})^2 - 1 = 9 e^{2i \theta} - 1.$ Hence, $\int_{0}^{2\pi} \frac{e^{3 e^{i \theta}}}{9 e^{2i \theta} - 1} \cdot 3i e^{i \theta} d\theta.$

3. Simplify the Integral: $\int_{0}^{2\pi} \frac{3i e^{i \theta} e^{3 e^{i \theta}}}{9 e^{2i \theta} - 1} d\theta.$

4. Recognize Symmetry and Poles: Notice that the poles of the integrand are at $z = \pm 1$. Since $|z| = 3$, both poles are inside the contour $|z| = 3$.

5. Residue Calculation (Without Using Residue Theorem): Even though we are not allowed to use the Residue Theorem directly, we can still use partial fraction decomposition to understand the residues. Let's decompose the integrand: $\frac{1}{z^2 - 1} = \frac{1}{(z-1)(z+1)}.$ So, $\frac{e^z}{z^2 - 1} = \frac{e^z}{(z-1)(z+1)}.$

6. Evaluate Each Part Separately: Let: $f(z) = \frac{e^z}{z+1} \quad \text{and} \quad g(z) = \frac{1}{z-1}.$

7. Integral of Each Part: Evaluate each part of the integral using parameterization: $\int_{|z|=3} f(z) dz = \int_{0}^{2\pi} \frac{e^{3 e^{i \theta}}}{3 e^{i \theta} + 1} 3i e^{i \theta} d\theta,$ $\int_{|z|=3} g(z) dz = \int_{0}^{2\pi} \frac{3i e^{i \theta}}{3 e^{i \theta} - 1} d\theta.$

8. Combining Results: Adding the integrals for $f(z)$ and $g(z)$: $\int_{|z|=3} \frac{e^z}{z^2 - 1} dz = \int_{0}^{2\pi} \left( \frac{e^{3 e^{i \theta}}}{3 e^{i \theta} + 1} + \frac{1}{3 e^{i \theta} - 1} \right) 3i e^{i \theta} d\theta.$

Since we're not supposed to use direct residues or Cauchy's formula, we'll rely on the parameterization and careful integral evaluation to solve this complex integral.

Would you like more detailed steps on any particular part of this solution?

Here are 8 related questions to consider:

1. What is the significance of parameterizing a contour integral in complex analysis?
2. How do you handle poles inside a contour when not using the Residue Theorem?
3. Can partial fraction decomposition always be applied to rational functions in complex analysis?
4. How does the parameterization method help in evaluating contour integrals?
5. What are the challenges of evaluating integrals with multiple poles inside the contour?
6. How does the orientation of the contour affect the integral in complex analysis?
7. What are other methods to evaluate complex integrals if residues are not allowed?
8. How can symmetry in the integrand simplify complex integrals?

Tip: Understanding the geometric interpretation of complex integrals can provide deeper insights into their evaluation and the effects of different contours.