https://r2.erweima.ai/mathbot/image/1719407863027z9xzx95m.png

To solve the given integral using Cauchy's integral formula, we start with the integral expression:

\[
\int_{\gamma} \frac{dz}{(z-1)(z-2)(z-3)}
\]

where \(\gamma\) is the contour \(|z| = 4\).

Cauchy's integral formula for a function \(f(z)\) is given by:

\[
f(a) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{z-a} dz
\]

For this problem, we will use the extended form of Cauchy's integral formula, known as the residue theorem, to solve the integral. The residue theorem states:

\[
\int_{\gamma} f(z) dz = 2\pi i \sum \text{Res}(f, a_i)
\]

where \(\text{Res}(f, a_i)\) are the residues of \(f(z)\) at its poles within the contour \(\gamma\).

The function given is:

\[
f(z) = \frac{1}{(z-1)(z-2)(z-3)}
\]

This function has poles at \(z = 1\), \(z = 2\), and \(z = 3\). We need to calculate the residues at each of these poles.

### Residue at \(z = 1\):

The residue at \(z = 1\) is:

\[
\text{Res}(f, 1) = \lim_{z \to 1} (z-1) f(z) = \lim_{z \to 1} \frac{1}{(z-2)(z-3)} = \frac{1}{(1-2)(1-3)} = \frac{1}{(-1)(-2)} = \frac{1}{2}
\]

### Residue at \(z = 2\):

The residue at \(z = 2\) is:

\[
\text{Res}(f, 2) = \lim_{z \to 2} (z-2) f(z) = \lim_{z \to 2} \frac{1}{(z-1)(z-3)} = \frac{1}{(2-1)(2-3)} = \frac{1}{(1)(-1)} = -1
\]

### Residue at \(z = 3\):

The residue at \(z = 3\) is:

\[
\text{Res}(f, 3) = \lim_{z \to 3} (z-3) f(z) = \lim_{z \to 3} \frac{1}{(z-1)(z-2)} = \frac{1}{(3-1)(3-2)} = \frac{1}{(2)(1)} = \frac{1}{2}
\]

### Sum of residues:

Now, we sum the residues:

\[
\sum \text{Res}(f, z_i) = \frac{1}{2} + (-1) + \frac{1}{2} = 0
\]

### Conclusion:

By the residue theorem, the integral is:

\[
\int_{\gamma} \frac{dz}{(z-1)(z-2)(z-3)} = 2\pi i \cdot 0 = 0
\]

Therefore, the value of the integral is \(0\).

Would you like further details or have any questions?

Here are some additional questions you might have:

1. How do you identify the poles of a complex function?
2. Can you explain the concept of residues in complex analysis?
3. What is Cauchy's integral formula, and how is it derived?
4. How does the residue theorem generalize Cauchy's integral formula?
5. Can you solve a different type of integral using Cauchy's integral formula?

**Tip:** Always verify the region of integration and ensure all poles are within the contour when applying the residue theorem.

Learn how to solve the complex integral ∫(gamma) dz / ((z-1)(z-2)(z-3)) using the Residue Theorem and Cauchy's Integral Formula. Detailed step-by-step solution with calculations of residues and application of key theorems in complex analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation