Cauchy's Integral Formula - Proof Sketch

Proof Sketch

By using the Cauchy integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around a. Since f(z) is continuous, we can choose a circle small enough on which f(z) is arbitrarily close to f(a). On the other hand, the integral

over any circle C centered at a. This can be calculated directly via a parametrization (integration by substitution) where 0 ≤ t ≤ 2π and ε is the radius of the circle.

Letting ε → 0 gives the desired estimate

\begin{align} \left | \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z-a} \,dz - f(a) \right | &= \left | \frac{1}{2 \pi i} \oint_C \frac{f(z)-f(a)}{z-a} \,dz \right |\\ &\leq \frac{1}{2 \pi} \int_0^{2\pi} \frac{ |f(z(t)) - f(a)| } {\varepsilon} \,\varepsilon\,dt\\ &\leq \max_{|z-a|=\varepsilon}|f(z) - f(a)| \xrightarrow{} 0. \end{align}

Read more about this topic:  Cauchy's Integral Formula

Famous quotes containing the words proof and/or sketch:

    “The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
    —Charles Baudelaire (1821–1867)

    “We criticize a man or a book most sharply when we sketch out their ideal.”
    —Friedrich Nietzsche (1844–1900)