The complex numbers ℂ have an extra symmetry: complex conjugation. Objects which respect the full structure of ℂ have to respect this symmetry, which leads to o

The complex numbers ℂ have an extra symmetry: complex conjugation. Objects which respect the full structure of ℂ have to respect this symmetry, which leads to overall nicer behavior. For example, if a complex function 𝑓 is differentiable once then... · It's differentiable an infinite number of times (smooth) · It has a Taylor series approximation at every point (analytic) · If γ is a smooth closed curve in the complex plane then the path integral over of 𝑓 around γ is 0, i.e., ∳ᵧ𝑓 = 0 (see https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem) · The real part of 𝑓 viewed as a function of ℝ² is harmonic (likewise for the imaginary part). This is a result of the Cauchy-Riemann equations: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equatio... · The residue theorem is very powerful and can be used to prove things about non-complex functions/integrals: https://en.wikipedia.org/wiki/Residue_theorem https://en.wikipedia.org/wiki/Residue_theorem The list goes on. Historically, complex analysis developed about 30 years before vector calculus as we know it today.