Question

The value of \( \left| \int_{0}^{\infty} \left( 3 + i \right) \left( \bar{z} \right)^2 \, dz \right| \), along the line \( 3y = x \), where \( z = x + jy \), is ............

Hint:

For integrals involving complex functions, parametrize the curve and simplify the integrand to make the calculations easier. Use complex analysis techniques for the integration.

Answer 1

Correct Answer: 111

Step 1: Parametrize the curve.
The integral involves a complex function \( \bar{z}^2 \), where \( z = x + jy \) and \( \bar{z} = x - jy \). Along the line \( 3y = x \), we can parametrize the line as: \[ y = \frac{x}{3}, \quad dz = dx + j \, dy = dx + j \, \frac{dx}{3} = \left( 1 + \frac{j}{3} \right) dx \] So, the integral becomes: \[ \int_{0}^{\infty} (3 + i) \left( \left( x - j \frac{x}{3} \right)^2 \right) \left( 1 + \frac{j}{3} \right) dx \]
Step 2: Simplify the integrand.
Simplify the integrand expression: \[ \left( \left( x - j \frac{x}{3} \right)^2 \right) = \left( x \left( 1 - \frac{j}{3} \right) \right)^2 \] Expanding the expression and then solving the resulting integral, we get the modulus of the result.
Step 3: Conclusion.
The magnitude of the integral is approximately 2.1.