Cauchy’s integral formula for derivatives

It’s soon exam time, so I’m practicing proofs in complex analysis. Right now that means Cauchy’s integral formula for \(n\)’th derivatives.

Let \(G\) be a domain of the complex numbers and \(f: G\to \mathbb{C}\) a holomorphic function. We first want to show that \(f\) can be expressed as a power series, such that $$f(z)=\sum^\infty_{n=0} a_n(z-a).$$ for some \(a\in\mathbb{C}\), let \(B_\rho(a)\) the largest open ball at \(a\) contained in \(G\). We claim that $$a_n = \frac{1}{2\pi i} \oint\frac{f(z)}{(z-a)^{n+1}}dz.$$ By the Cauchy integral formula we have that, for a fixed \(z_0\in B_\rho(a)\) we have $$f(z_0)=\frac{1}{2\pi i} \oint \frac{f(z)}{z-z_0}$$ and by elementary calculations we can, for \(z\in \partial B_r(a)\), write $$\frac{1}{z-z_0} = \frac{1}{z-a} \frac{1}{1-\frac{z_0 -a}{z-a}}=\frac{1}{z-a}\sum^\infty_{n=0} \left(\frac{z_0-a}{z-a}\right)^n,$$ and from above we then have $$\begin{split} f(z_0)& = \frac{1}{2\pi i} \oint \frac{f(z)}{z-z_0} \\ &=\frac{1}{2\pi i}\oint\sum^\infty_{n=0} \frac{f(z)(z_0-a)^n}{(z-a)^{n+1}}dz\\ &=\frac{1}{2\pi i}\sum^\infty_{n=0}\oint \frac{f(z)(z_0-a)^n}{(z-a)^{n+1}}dz\\&=\frac{1}{2\pi i}\sum^\infty_{n=0}\oint \frac{f(z)}{(z-a)^{n+1}}dz(z_0-a)^n\\ &=\sum^\infty_{n=0}a_n (z_0-a)^n,\end{split}$$ as wanted. We see that \(f\) is a power series and thus infinitely complex differentiable, and the derivatives are $$f^{(n)}(a)=\frac{n!}{2\pi i}\oint\frac{f(z)}{(z-a)^{n+1}}dz,$$ as desired.

The observant reader will have noticed that I didn’t check that the sums were uniformly convergent, which is needed in other to switch the sum and integral signs, but this is an easy application of the Weierstraß \(M\)-test.

References

Berg, C. (2016). Complex analysis. Copenhagen: Department of Mathematical Sciences, University of Copenhagen.

4 thoughts on “Cauchy’s integral formula for derivatives

    • Hi! Thanks for your comment 🙂 This result isn’t new though, this formula has been known for a very long time. Any book about complex analysis should give you much more detail.

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