In complex analysis, all differentiable (in a region) functions are infinitely differentiable, and are the same as their Taylor series.
Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down.
I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like, "We draw a path, then calculate this contour integral, and magic happens."
Fundamental Theorem of Algebra: Every nonconstant polynomial over the complex numbers has a complex root.
Proof: Suppose that p(z) is a polynomial over the complex numbers with no root.
Consider the function 1/p(z). If p(z) had no roots, then 1/p(z) is entire. But we can bound it for everything outside of a large circle because the leading term dominates the others. And since the large circle is compact, we can bound p(z) away from 0 inside the circle. Between the two, 1/p(z) is bounded, and so much be constant by Liouville's theorem.
But 1/p(z) is only constant if p(z) is constant. Therefore any complex polynomial with no complex roots must be constant.
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I'm convinced by the proof. But part of me still says that it is magic.
And the definite integrals you can get out of the Cauchy integral formula are awesome.
I think it is because being holomorphic is so much stronger even than C^\infty (continuous with all derivatives continuous) real much less just continuous or merely integrable.
Instead of thinking of real analysis as a zoo of weird exceptions, it's probably more accurate to think of complex analyic functions as the exceptions. For example, when viewed as a two dimensional mapping from the plane to itself, complex-analytic functions are conformal (angle-preserving) whereas most differentiable mappings from the plane to itself are not.
On the other hand, complex analysis explains things that aren’t obvious in real analysis, like why the radius of convergence for a continuous function on the real line might be 1 (turns out there are singularities off the real line on the complex plane).
Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down.
I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like, "We draw a path, then calculate this contour integral, and magic happens."