Analyticity of Holomorphic Functions - Proof

Proof

The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series development of the expression

.

Suppose ƒ is differentiable everywhere within some open disk centered at a. Let z be within that open disk. Let C be a positively oriented (i.e., counterclockwise) circle centered at a, lying within that open disk but farther from a than z is. Starting with Cauchy's integral formula, we have

\begin{align}f(z) &{}= {1 \over 2\pi i}\int_C {f(w) \over w-z}\,\mathrm{d}w \\ &{}= {1 \over 2\pi i}\int_C {f(w) \over (w-a)-(z-a)} \,\mathrm{d}w \\ &{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{1 \over 1-{z-a \over w-a}}f(w)\,\mathrm{d}w \\ &{}={1 \over 2\pi i}\int_C {1 \over w-a}\cdot{\sum_{n=0}^\infty\left({z-a \over w-a}\right)^n} f(w)\,\mathrm{d}w \\ &{}=\sum_{n=0}^\infty{1 \over 2\pi i}\int_C {(z-a)^n \over (w-a)^{n+1}} f(w)\,\mathrm{d}w.\end{align}

To justify the interchange of the sum and the integral, one must notice that in the intersection of |(z − a)/(w − a)| ≤ r < 1 and some closed domain containing C, ƒ(w)/(w − a) is holomorphic and therefore bounded by some positive number M. So we have

The Weierstrass M-test says the series converges uniformly, and thus the interchange of the sum and the integral is justified.

Since the factor (z − a)n does not depend on the variable of integration w, it can be pulled out:

And now the integral and the factor of 1/(2πi) do not depend on z, i.e., as a function of z, that whole expression is a constant cn, so we can write:

and that is the desired power series.

Read more about this topic:  Analyticity Of Holomorphic Functions

Famous quotes containing the word proof:

    “The moment a man begins to talk about technique that’s proof that he is fresh out of ideas.”
    —Raymond Chandler (1888–1959)

    “Right and proof are two crutches for everything bent and crooked that limps along.”
    —Franz Grillparzer (1791–1872)

    “If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
    —Herman Melville (1819–1891)