4.1: Introduction to Line Integrals and Cauchy’s Theorem
- Page ID
- 6480
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The basic theme here is that complex line integrals will mirror much of what we’ve seen for multi- variable calculus line integrals. But, just like working with \(e^{i \theta}\) is easier than working with sine and cosine, complex line integrals are easier to work with than their multivariable analogs. At the same time they will give deep insight into the workings of these integrals.
To define complex line integrals, we will need the following ingredients:
- The complex plane: \(z = x + iy\)
- The complex differential \(dz = dx + idy\)
- A curve in the complex plane: \(\gamma (t) = x(t) + iy(t)\), defined for \(a \le t \le b\).
- A complex function: \(f(z) = u(x, y) + iv(x, y)\)