The question asks to evaluate the contour integral $\int_C \overline{z} \, dz$ along a specific path $C$, which is the line segment from the origin $0$ to $1 + i$.

Solution

1. Parameterize the Path $C$: The line segment from $0$ to $1 + i$ can be parameterized by setting $z(t) = t + it$, where $t$ goes from $0$ to $1$.

2. Express $dz$: Taking the derivative with respect to $t$, we get: $\frac{dz}{dt} = 1 + i \Rightarrow dz = (1 + i) \, dt$

3. Express $\overline{z}$: The complex conjugate of $z$ along this path is $\overline{z} = t - it$.

4. Set Up the Integral: Substitute $\overline{z} = t - it$ and $dz = (1 + i) \, dt$ into the integral: $\int_C \overline{z} \, dz = \int_0^1 (t - it)(1 + i) \, dt$

5. Simplify the Integrand: Expanding $(t - it)(1 + i)$: $(t - it)(1 + i) = t + it - it - i^2 t = t + t = 2t$ So, the integral simplifies to: $\int_0^1 2t \, dt$

6. Evaluate the Integral: Now we have a simple real integral: $\int_0^1 2t \, dt = 2 \int_0^1 t \, dt = 2 \left[ \frac{t^2}{2} \right]_0^1 = 2 \cdot \frac{1}{2} = 1$

Answer

The value of the contour integral is: $\int_C \overline{z} \, dz = 1$

Would you like more details on any part of this solution, or do you have further questions?

Related Questions:

1. How would the solution change if the path $C$ were different, such as a circular arc?
2. Can we use this method for a general complex function, or does it work only for certain types?
3. What is the role of conjugation in complex integration, and how does it affect the result?
4. How do contour integrals apply in the context of complex analysis?
5. What are some applications of complex integration in physics and engineering?

Tip

When solving contour integrals involving conjugates, make sure to parameterize correctly and carefully handle the conjugation within the integrand.