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:

    “If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
    —Thomas Jefferson (1743–1826)

    “the vagabond began
    To sketch a face that well might buy the soul of any man.
    Then, as he placed another lock upon the shapely head,
    With a fearful shriek, he leaped and fell across the
    picture—dead.”
    —Hugh Antoine D’Arcy (1843–1925)