Math Problem Statement

Solution

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

γdz(z1)(z2)(z3)\int_{\gamma} \frac{dz}{(z-1)(z-2)(z-3)}

where γ\gamma is the contour z=4|z| = 4.

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

f(a)=12πiγf(z)zadzf(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:

γf(z)dz=2πiRes(f,ai)\int_{\gamma} f(z) dz = 2\pi i \sum \text{Res}(f, a_i)

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

The function given is:

f(z)=1(z1)(z2)(z3)f(z) = \frac{1}{(z-1)(z-2)(z-3)}

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

Residue at z=1z = 1:

The residue at z=1z = 1 is:

Res(f,1)=limz1(z1)f(z)=limz11(z2)(z3)=1(12)(13)=1(1)(2)=12\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=2z = 2:

The residue at z=2z = 2 is:

Res(f,2)=limz2(z2)f(z)=limz21(z1)(z3)=1(21)(23)=1(1)(1)=1\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=3z = 3:

The residue at z=3z = 3 is:

Res(f,3)=limz3(z3)f(z)=limz31(z1)(z2)=1(31)(32)=1(2)(1)=12\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:

Res(f,zi)=12+(1)+12=0\sum \text{Res}(f, z_i) = \frac{1}{2} + (-1) + \frac{1}{2} = 0

Conclusion:

By the residue theorem, the integral is:

γdz(z1)(z2)(z3)=2πi0=0\int_{\gamma} \frac{dz}{(z-1)(z-2)(z-3)} = 2\pi i \cdot 0 = 0

Therefore, the value of the integral is 00.

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.

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Residue Theorem
Cauchy's Integral Formula

Formulas

Cauchy's Integral Formula
Residue Theorem

Theorems

Residue Theorem

Suitable Grade Level

Advanced Undergraduate