Contour Integral of Real Part Re(z) | Direct Parametrization Example

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In this video, we evaluate a complex contour integral involving the real part of a function along a parabolic path.

We solve the integral:
I = ∫ Re(z + z²) dz
along the parabola y = 2x² from x=0 to x=1.

Since the integrand involves the Real part operator Re(z), the function is not analytic, making the integral path-dependent. We solve it using Direct Parametrization.

Step-by-step solution visualized with Manim:
1. Define the path: z = x + i(2x²).
2. Simplify the integrand Re(z + z²) in terms of x and y.
3. Compute the differential dz = (1 + 4xi) dx.
4. Substitute everything into the integral to get a definite integral in terms of x.
5. Evaluate the integral to find the final complex number: 1/30(1 - 10i).

This example demonstrates how to handle non-analytic functions in Complex Analysis.
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Видео Contour Integral of Real Part Re(z) | Direct Parametrization Example канала Math Infinitum