To solve the given integral using complex analysis, we will follow the steps: factorization, finding critical points, and verifying that the integrals are equal. The integrals given are:

$\int_{|z|=1} \frac{dz}{z^3(z^2+10)} = \int_{|z|=2} \frac{dz}{z^3(z^2+10)}$

Step 1: Factorize the Denominator

First, let's rewrite the integrand:

$\frac{1}{z^3(z^2 + 10)}$

Step 2: Find the Critical Points

The critical points (poles) of the integrand occur where the denominator is zero. The denominator is $z^3(z^2 + 10)$, which factors into:

$z^3(z^2 + 10) = 0$

The roots are:

• $z = 0$
• $z^2 + 10 = 0$ which gives $z = \pm i\sqrt{10}$

So the poles are at $z = 0$ and $z = \pm i\sqrt{10}$.

Step 3: Evaluate the Contour Integrals

We need to show that the contour integrals over $|z|=1$ and $|z|=2$ are equal. By the Residue Theorem, if the poles of the integrand are not enclosed by the contours, the integrals over these contours will be equal.

For the contour $|z|=1$:

• The poles $z = 0$ and $z = \pm i\sqrt{10}$ are outside the contour since $|\pm i\sqrt{10}| = \sqrt{10} > 1$.

For the contour $|z|=2$:

• The poles $z = 0$ and $z = \pm i\sqrt{10}$ are inside the contour since $\sqrt{10} < 2$.

The Residue Theorem states that the integral of a function around a closed contour is $2\pi i$ times the sum of the residues of the function inside the contour. Since the integrand $\frac{1}{z^3(z^2 + 10)}$ has no poles inside $|z|=1$, its integral over $|z|=1$ is zero. The integral over $|z|=2$ is also zero because the sum of residues at the poles inside $|z|=2$ is zero (considering the sum of residues at $z=0$ and $z = \pm i\sqrt{10}$ equals zero).

Hence, the integrals are equal:

$\int_{|z|=1} \frac{dz}{z^3(z^2 + 10)} = \int_{|z|=2} \frac{dz}{z^3(z^2 + 10)} = 0$

Conclusion

We have shown that the integrals over the contours $|z|=1$ and $|z|=2$ are indeed equal by using the Residue Theorem.

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Would you like any additional details or have any questions? Here are some related questions for further exploration:

1. What are the conditions for the Residue Theorem to be applicable?
2. How do you find residues at simple poles?
3. How do you compute integrals involving higher-order poles?
4. Can you explain the significance of the Cauchy-Goursat theorem?
5. How would the problem change if there were poles inside both contours?
6. What is the relationship between contour integration and real integrals?
7. Can you provide an example of a contour integral with non-zero residues?
8. How does the choice of contour affect the evaluation of an integral?

Tip: When dealing with complex integrals, always check if the poles lie within the given contour to effectively use the Residue Theorem.