Cauchy's integral formula

Proof sketch

By using the Cauchy integral theorem, one can show that the integral over ${\displaystyle C}$ (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around ⁠${\displaystyle a}$⁠. Since ${\displaystyle f(z)}$ is continuous, we can choose a circle small enough on which ${\displaystyle f(z)}$ is arbitrarily close to ⁠${\displaystyle f(a)}$⁠. On the other hand, the integral $${\displaystyle {\oint }_C\frac{1}{z-a}dz=2\pi i,}$$ over any circle ${\displaystyle C}$ centered at ⁠${\displaystyle a}$⁠. This can be calculated directly via a parametrization (integration by substitution) ${\displaystyle z(t)=a+\epsilon e^{it}}$ where ${\displaystyle 0\le t\le 2\pi }$ and ${\displaystyle \epsilon }$ is the radius of the circle.

Letting ${\displaystyle \epsilon \to 0}$ gives the desired estimate $${\displaystyle \begin{array}{cc}\left|\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{z-a}dz-f(a)\right|& =\left|\frac{1}{2\pi i}{\oint }_C\frac{f(z)-f(a)}{z-a}dz\right|\\ & =\left|\frac{1}{2\pi i}{\int }_0^{2\pi }\left(\frac{f(z(t))-f(a)}{\epsilon e^{it}}\cdot \epsilon e^{it}i\right)dt\right|\\ & \le \frac{1}{2\pi }{\int }_0^{2\pi }\frac{\left|f(z(t))-f(a)\right|}{\epsilon }\epsilon dt\\ & \le \underset{|z-a|=\epsilon }{max}\left|f(z)-f(a)\right|\underset{\epsilon \to 0}{\overset{}{\to }}0.\end{array}}$$

Example

Let $${\displaystyle g(z)=\frac{z^2}{z^2+2z+2},}$$ and let ${\displaystyle C}$ be the contour described by ${\displaystyle |z|=2}$ (the circle of radius 2).

To find the integral of ${\displaystyle g(z)}$ around the contour ⁠${\displaystyle C}$⁠, we need to know the singularities of ⁠${\displaystyle g(z)}$⁠. Observe that we can rewrite ${\displaystyle g}$ as follows: $${\displaystyle g(z)=\frac{z^2}{(z-z_1)(z-z_2)}}$$ where ${\displaystyle z_1=-1+i}$ and ⁠${\displaystyle z_2=-1-i}$⁠.

Thus, ${\displaystyle g}$ has poles at ${\displaystyle z_1}$ and ⁠${\displaystyle z_2}$⁠. The moduli of these points are less than 2 and thus lie inside the contour. This integral can be split into two smaller integrals by Cauchy–Goursat theorem; that is, we can express the integral around the contour as the sum of the integral around ${\displaystyle z_1}$ and ${\displaystyle z_2}$ where the contour is a small circle around each pole. Call these contours ${\displaystyle C_1}$ around ${\displaystyle z_1}$ and ${\displaystyle C_2}$ around ⁠${\displaystyle z_2}$⁠.

Now, each of these smaller integrals can be evaluated by the Cauchy integral formula, but they first must be rewritten to apply the theorem. For the integral around ⁠${\displaystyle C_1}$⁠, define ${\displaystyle f_1}$ as ⁠${\displaystyle f_1(z)=(z-z_1)g(z)}$⁠. This is analytic (since the contour does not contain the other singularity). We can simplify ${\displaystyle f_1}$ to be: $${\displaystyle f_1(z)=\frac{z^2}{z-z_2}}$$ and now $${\displaystyle g(z)=\frac{f_1(z)}{z-z_1}.}$$

Since the Cauchy integral formula says that: $${\displaystyle {\oint }_C\frac{f_1(z)}{z-a}dz=2\pi i\cdot f_1(a),}$$ we can evaluate the integral as follows: $${\displaystyle {\oint }_{C_1}g(z)dz={\oint }_{C_1}\frac{f_1(z)}{z-z_1}dz=2\pi i\frac{z_1^2}{z_1-z_2}.}$$

Doing likewise for the other contour: $${\displaystyle f_2(z)=\frac{z^2}{z-z_1},}$$ we evaluate $${\displaystyle {\oint }_{C_2}g(z)dz={\oint }_{C_2}\frac{f_2(z)}{z-z_2}dz=2\pi i\frac{z_2^2}{z_2-z_1}.}$$

The integral around the original contour ${\displaystyle C}$ then is the sum of these two integrals: $${\displaystyle \begin{array}{cc}{\oint }_{C}g(z)dz& ={\oint }_{C_1}g(z)dz+{\oint }_{C_2}g(z)dz\\ & =2\pi i\left(\frac{z_1^2}{z_1-z_2}+\frac{z_2^2}{z_2-z_1}\right)\\ & =2\pi i(-2)\\ & =-4\pi i.\end{array}}$$

An elementary trick using partial fraction decomposition: $${\displaystyle {\oint }_{C}g(z)dz={\oint }_C\left(1-\frac{1}{z-z_1}-\frac{1}{z-z_2}\right)dz=0-2\pi i-2\pi i=-4\pi i}$$

Consequences

The integral formula has broad applications. First, it implies that a function that is holomorphic in an open set is in fact infinitely differentiable there. Furthermore, it is an analytic function, meaning that it can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series applied to $${\displaystyle f(\zeta )=\frac{1}{2\pi i}{\int }_C\frac{f(z)}{z-\zeta }dz.}$$

The formula is also used to prove the residue theorem, which is a result for meromorphic functions, and a related result, the argument principle. It is known from Morera's theorem that the uniform limit of holomorphic functions is holomorphic. This can also be deduced from Cauchy's integral formula: indeed the formula also holds in the limit and the integrand, and hence the integral, can be expanded as a power series. In addition the Cauchy formulas for the higher order derivatives show that all these derivatives also converge uniformly.

The analog of the Cauchy integral formula in real analysis is the Poisson integral formula for harmonic functions; many of the results for holomorphic functions carry over to this setting. No such results, however, are valid for more general classes of differentiable or real analytic functions. For instance, the existence of the first derivative of a real function need not imply the existence of higher order derivatives, nor in particular the analyticity of the function. Likewise, the uniform limit of a sequence of (real) differentiable functions may fail to be differentiable, or may be differentiable but with a derivative that is not the limit of the derivatives of the members of the sequence.

Another consequence is that if ${\displaystyle f(z)=\sum a_n{z}^n}$ is holomorphic in ${\displaystyle |z|<R}$ and ${\displaystyle 0<r<R}$ then the coefficients ${\displaystyle a_n}$ satisfy Cauchy's estimate $${\displaystyle |a_n|=|\underset{=a_n}{\underbrace{\frac{1}{2\pi i}{\int }_C\frac{f(z)}{(z-\zeta {)}^{n+1}}dz}}|\le r^{-n}\underset{|z|=r}{sup}|f(z)|.}$$

From Cauchy's estimate, one can easily deduce that every bounded entire function must be constant (which is Liouville's theorem).

The formula can also be used to derive Gauss's mean-value theorem, which states $${\displaystyle f(z)=\frac{1}{2\pi }{\int }_0^{2\pi }f(z+r{e}^{i\theta })d\theta .}$$

In other words, the average value of ${\displaystyle f}$ over the circle centered at ${\displaystyle z}$ with radius ${\displaystyle r}$ is ⁠${\displaystyle f(z)}$⁠. This can be calculated directly via a parametrization of the circle.

Generalizations

Smooth functions

A version of Cauchy's integral formula is the Cauchy–Pompeiu formula, and holds for smooth functions as well, as it is based on Stokes' theorem. Let ${\displaystyle D}$ be a disc in ${\displaystyle \mathbb{C}}$ and suppose that ${\displaystyle f}$ is a complex-valued C¹ function on the closure of ⁠${\displaystyle D}$⁠. Then $${\displaystyle f(\zeta )=\frac{1}{2\pi i}{\int }_{\mathrm{\partial }D}\frac{f(z)dz}{z-\zeta }-\frac{1}{\pi }{\iint }_D\frac{\mathrm{\partial }f}{\mathrm{\partial }\overline{z}}(z)\frac{dx\wedge dy}{z-\zeta }.}$$

One may use this representation formula to solve the inhomogeneous Cauchy–Riemann equations in ⁠${\displaystyle D}$⁠. Indeed, if ${\displaystyle \phi }$ is a function in ⁠${\displaystyle D}$⁠, then a particular solution ${\displaystyle f}$ of the equation is a holomorphic function outside the support of ⁠${\displaystyle \mu }$⁠. Moreover, if in an open set ⁠${\displaystyle D}$⁠, $${\displaystyle d\mu =\frac{1}{2\pi i}\phi dz\wedge d\overline{z}}$$ for some ${\displaystyle \phi \in C^k(D)}$ (where ⁠${\displaystyle k\ge 1}$⁠, then ${\displaystyle f(\zeta ,\overline{\zeta })}$ is also in ${\displaystyle C^k(D)}$ and satisfies the equation $${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\overline{z}}=\phi (z,\overline{z}).}$$

The first conclusion is, succinctly, that the convolution ${\displaystyle \mu \ast k(z)}$ of a compactly supported measure with the Cauchy kernel $${\displaystyle k(z)=\mathrm{p}.\mathrm{v}.\frac{1}{z}}$$ is a holomorphic function off the support of ⁠${\displaystyle \mu }$⁠. Here ${\displaystyle \mathrm{p}.\mathrm{v}.}$ denotes the principal value. The second conclusion asserts that the Cauchy kernel is a fundamental solution of the Cauchy–Riemann equations. Note that for smooth complex-valued functions ${\displaystyle f}$ of compact support on ${\displaystyle \mathbb{C}}$ the generalized Cauchy integral formula simplifies to $${\displaystyle f(\zeta )=\frac{1}{2\pi i}\iint \frac{\mathrm{\partial }f}{\mathrm{\partial }\overline{z}}\frac{dz\wedge d\overline{z}}{z-\zeta },}$$ and is a restatement of the fact that, considered as a distribution, ${\displaystyle (\pi z{)}^{-1}}$ is a fundamental solution of the Cauchy–Riemann operator ⁠${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}$⁠.

The generalized Cauchy integral formula can be deduced for any bounded open region ${\displaystyle X}$ with ${\displaystyle C^1}$ boundary ${\displaystyle \mathrm{\partial }X}$ from this result and the formula for the distributional derivative of the characteristic function ${\displaystyle {\chi }_X}$ of ⁠${\displaystyle X}$⁠: $${\displaystyle \frac{\mathrm{\partial }{\chi }_X}{\mathrm{\partial }\overline{z}}=\frac{i}{2}{\oint }_{\mathrm{\partial }X}dz,}$$ where the distribution on the right hand side denotes contour integration along ⁠${\displaystyle \mathrm{\partial }X}$⁠.

Proof

For ${\displaystyle \phi \in \mathcal{D}(X)}$ calculate: $${\displaystyle \begin{array}{cc}⟨\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left({\chi }_X\right),\phi ⟩& =-{\int }_X\frac{\mathrm{\partial }\phi }{\mathrm{\partial }\overline{z}}\mathrm{d}(x,y)\\ & =-\frac{1}{2}{\int }_X\left({\mathrm{\partial }}_x\phi +\mathrm{i}{\mathrm{\partial }}_y\phi \right)\mathrm{d}(x,y).\end{array}}$$ then traverse ${\displaystyle \mathrm{\partial }X}$ in the anti-clockwise direction. Fix a point ${\displaystyle p\in \mathrm{\partial }X}$ and let ${\displaystyle s}$ denote arc length on ${\displaystyle \mathrm{\partial }X}$ measured from ${\displaystyle p}$ anti-clockwise. Then, if ${\displaystyle \ell }$ is the length of ${\displaystyle \mathrm{\partial }X,[0,\ell ]\ni s\mapsto (x(s),y(s))}$ is a parametrization of ⁠${\displaystyle \mathrm{\partial }X}$⁠. The derivative ${\displaystyle \tau =\left(x^{\prime }(s),y^{\prime }(s)\right)}$ is a unit tangent to ${\displaystyle \mathrm{\partial }X}$ and ${\displaystyle \nu :=\left(-y^{\prime }(s),x^{\prime }(s)\right)}$ is the unit outward normal on ⁠${\displaystyle \mathrm{\partial }X}$⁠. We are lined up for use of the divergence theorem: put ${\displaystyle V=(\phi ,\mathrm{i}\phi )\in \mathcal{D}(X{)}^2}$ so that ${\displaystyle \mathrm{div}V={\mathrm{\partial }}_x\phi +\mathrm{i}{\mathrm{\partial }}_y\phi }$ and we get $${\displaystyle \begin{array}{cc}-\frac{1}{2}{\int }_X\left({\mathrm{\partial }}_x\phi +\mathrm{i}{\mathrm{\partial }}_y\phi \right)\mathrm{d}(x,y)& =-\frac{1}{2}{\int }_{\mathrm{\partial }X}V\cdot \nu \mathrm{d}S\\ & =-\frac{1}{2}{\int }_0^{\ell }\left(\phi {\nu }_1+\mathrm{i}\phi {\nu }_2\right)\mathrm{d}s\\ & =-\frac{1}{2}{\int }_0^{\ell }\phi (x(s),y(s))\left(y^{\prime }(s)-\mathrm{i}{x}^{\prime }(s)\right)\mathrm{d}s\\ & =\frac{1}{2}{\int }_0^{\ell }\mathrm{i}\phi (x(s),y(s))\left(x^{\prime }(s)+\mathrm{i}{y}^{\prime }(s)\right)\mathrm{d}s\\ & =\frac{\mathrm{i}}{2}{\int }_{\mathrm{\partial }X}\phi \mathrm{d}z\end{array}}$$

Hence we proved ⁠${\displaystyle 1=\frac{\mathrm{\partial }{\chi }_X}{\mathrm{\partial }\overline{z}}=\frac{i}{2}{\oint }_{\mathrm{\partial }X}dz}$⁠.

Now we can deduce the generalized Cauchy integral formula:

Proof

Since $u=\frac{{\chi }_X}{\pi \left(z-z_0\right)}\in {\mathrm{L}}_{\text{loc}}^1(X)$ and since ${\displaystyle z_0\in X}$ this distribution is locally in ${\displaystyle X}$ of the form "distribution times ${\displaystyle C^{\mathrm{\infty }}}$ function", so we may apply the Leibniz rule to calculate its derivatives: $${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\overline{z}}=\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left(\frac{1}{\pi \left(z-z_0\right)}\right){\chi }_X+\frac{1}{\pi \left(z-z_0\right)}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left({\chi }_X\right)}$$

Using that ${\displaystyle (\pi z{)}^{-1}}$ is a fundamental solution of the Cauchy–Riemann operator ⁠${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}$⁠, we get ⁠${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left(\frac{1}{\pi \left(z-z_0\right)}\right)={\delta }_{z_0}}$⁠: $${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\overline{z}}={\delta }_{z_0}+\frac{1}{\pi \left(z-z_0\right)}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left({\chi }_X\right)}$$

Applying $\frac{\mathrm{\partial }u}{\mathrm{\partial }\overline{z}}$ to ⁠${\displaystyle \varphi \in \mathcal{D}(X)}$⁠: $${\displaystyle \begin{array}{cc}⟨\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left(\frac{{\chi }_X}{\pi \left(z-z_0\right)}\right),\varphi ⟩& =\varphi \left(z_0\right)+⟨\frac{1}{\pi \left(z-z_0\right)}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left({\chi }_X\right),\varphi ⟩\\ & =\varphi \left(z_0\right)+⟨\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\left({\chi }_X\right),\frac{\varphi }{\pi \left(z-z_0\right)}⟩\\ & =\varphi \left(z_0\right)+\frac{\mathrm{i}}{2}{\int }_{\mathrm{\partial }X}\frac{\varphi (z)}{\pi \left(z-z_0\right)}\mathrm{d}z\end{array}}$$ where $\frac{\mathrm{\partial }{\chi }_X}{\mathrm{\partial }\overline{z}}=\frac{i}{2}{\oint }_{\mathrm{\partial }X}dz$ is used in the last line.

Rearranging, we get $${\displaystyle \varphi (z_0)=\frac{1}{2\pi i}{\int }_{\mathrm{\partial }X}\frac{\varphi (z)dz}{z-z_0}-\frac{1}{\pi }{\iint }_X\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\overline{z}}(z)\frac{dx\wedge dy}{z-z_0}.}$$ as desired.

Several variables

In several complex variables, the Cauchy integral formula can be generalized to polydiscs. Let ${\displaystyle D}$ be the polydisc given as the Cartesian product of ${\displaystyle n}$ open discs ⁠${\displaystyle D_1,\dots ,D_n}$⁠: $${\displaystyle D=\prod _{j=1}^n{D}_j.}$$

Suppose that ${\displaystyle f}$ is a holomorphic function in ${\displaystyle D}$ continuous on the closure of ⁠${\displaystyle D}$⁠. Then $${\displaystyle f(\zeta )=\frac{1}{{\left(2\pi i\right)}^n}\int \cdots {\iint }_{\mathrm{\partial }{D}_1\times \cdots \times \mathrm{\partial }{D}_n}\frac{f(z_1,\dots ,z_n)}{(z_1-{\zeta }_1)\cdots (z_n-{\zeta }_n)}d{z}_1\cdots d{z}_n}$$ where ⁠${\displaystyle \zeta =({\zeta }_1,\dots ,{\zeta }_n)\in D}$⁠.

In real algebras

The Cauchy integral formula is generalizable to real vector spaces of two or more dimensions. The insight into this property comes from geometric algebra, where objects beyond scalars and vectors (such as planar bivectors and volumetric trivectors) are considered, and a proper generalization of Stokes' theorem.

Geometric calculus defines a derivative operator ${\displaystyle \mathrm{\nabla }={\hat{e}}_j{\mathrm{\partial }}_j}$ under its geometric product – that is, for a ${\displaystyle k}$-vector field ⁠${\displaystyle \psi (r)}$⁠, the derivative ${\displaystyle \mathrm{\nabla }\psi }$ generally contains terms of grade ${\displaystyle k+1}$ and ⁠${\displaystyle k-1}$⁠. For example, a vector field ${\displaystyle k=1}$ generally has in its derivative a scalar part, the divergence (⁠${\displaystyle k=0}$⁠), and a bivector part, the curl (⁠${\displaystyle k=2}$⁠). This particular derivative operator has a Green's function: $${\displaystyle G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)=\frac{1}{S_n}\frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}^n}}$$ where ${\displaystyle S_n}$ is the surface area of a unit ${\displaystyle n}$-ball in the space (that is, ⁠${\displaystyle S_2=2\pi }$⁠, the circumference of a circle with radius ⁠${\displaystyle 1}$⁠, and ⁠${\displaystyle S_3=4\pi }$⁠, the surface area of a sphere with radius ⁠${\displaystyle 1}$⁠). By definition of a Green's function, $${\displaystyle \mathrm{\nabla }G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)=\delta \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right).}$$

It is this useful property that can be used, in conjunction with the generalized Stokes theorem: $${\displaystyle {\oint }_{\mathrm{\partial }V}d\mathbf{S}f(\mathbf{r})={\int }_{V}d\mathbf{V}\mathrm{\nabla }f(\mathbf{r})}$$ where, for an ${\displaystyle n}$-dimensional vector space, ${\displaystyle dS}$ is an ${\displaystyle n-1}$-vector and ${\displaystyle dV}$ is an ${\displaystyle n}$-vector. The function ${\displaystyle f(r)}$ can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using the generalized Stokes theorem on the quantity ${\displaystyle G(r,r^{\prime })f(r)}$ and use of the product rule: $${\displaystyle {\oint }_{\mathrm{\partial }{V}^{\prime }}G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)d{\mathbf{S}}^{\prime }f\left({\mathbf{r}}^{\prime }\right)={\int }_V\left(\left[{\mathrm{\nabla }}^{\prime }G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)\right]f\left({\mathbf{r}}^{\prime }\right)+G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right){\mathrm{\nabla }}^{\prime }f\left({\mathbf{r}}^{\prime }\right)\right)d\mathbf{V}}$$

When ⁠${\displaystyle \mathrm{\nabla }f=0}$⁠, ${\displaystyle f(r)}$ is called a monogenic function, the generalization of holomorphic functions to higher-dimensional spaces – indeed, it can be shown that the Cauchy–Riemann condition is just the two-dimensional expression of the monogenic condition. When that condition is met, the second term in the right-hand integral vanishes, leaving only $${\displaystyle {\oint }_{\mathrm{\partial }{V}^{\prime }}G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)d{\mathbf{S}}^{\prime }f\left({\mathbf{r}}^{\prime }\right)={\int }_V\left[{\mathrm{\nabla }}^{\prime }G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)\right]f\left({\mathbf{r}}^{\prime }\right)=-{\int }_V\delta \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)f\left({\mathbf{r}}^{\prime }\right)d\mathbf{V}=-i_{n}f(\mathbf{r})}$$ where ${\displaystyle i_n}$ is that algebra's unit ${\displaystyle n}$-vector, the pseudoscalar. The result is $${\displaystyle f(\mathbf{r})=-\frac{1}{i_n}{\oint }_{\mathrm{\partial }V}G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)d\mathbf{S}f\left({\mathbf{r}}^{\prime }\right)=-\frac{1}{i_n}{\oint }_{\mathrm{\partial }V}\frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{S_n{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}^n}d\mathbf{S}f\left({\mathbf{r}}^{\prime }\right)}$$

Thus, as in the two-dimensional (complex analysis) case, the value of an analytic (monogenic) function at a point can be found by an integral over the surface surrounding the point, and this is valid not only for scalar functions but vector and general multivector functions as well.