• Complex Analysis: #5 Cauchy`s Integral Formula

Theorem 6
Let G ⊂ ℂ be a region and let f : G → ℂ be analytic. Take z₀ ∈ G and r > 0 so small that {z ∈ ℂ : |z − z₀| ≤ r} ⊂ G. Furthermore, let |a − z₀| < r. Then

converges to the constant number f '(a) in the limit as ∈ → 0. On the other hand, the path length around the circle, and the tangent vector to this path, approach zero as ∈ → 0. Thus in the limit, the first integral is zero. As far as the second integral is concerned, we have

A relatively trivial implication is the following theorem.


Theorem 7
The same assumptions as in Theorem 6. But this time take z₀ to be the central point of the circle. Then

So this is just a kind of “mean value theorem” for analytic functions. It shows quite clearly the difference between real analysis and complex analysis. In real analysis, we can make a smooth change in a function, leaving everything far away unchanged, and the function remains nicely differentiable. But in complex analysis, the precise value of the function is determined by the values on a circle, perhaps far away from the point we are looking at. So a change at one place implies that the whole function must change everywhere in order to remain analytic.