Cauchy's Integral Theorem and Formula

• Cauchy's Integral Theorem: ∮_C f(z)dz = 0 for holomorphic f on closed curve C; reveals rigidity of complex analytic functions.
• Cauchy's Integral Formula: f(z₀) = (1/2πi)∮_C f(z)/(z-z₀)dz determines function values inside from boundary data alone.
• Holomorphic functions are infinitely differentiable and analytic; complex differentiability is far stronger than real differentiability.
• Foundational for residue theorem, maximum modulus principle, and computing difficult real integrals via complex analysis.
• Modern rigorous formulation developed through 1800s by Riemann, Weierstrass, and Goursat (eliminated need for derivative continuity).

Cauchy's Integral Theorem and Formula

Overview

The importance of these theorems cannot be overstated. They provide the foundation for the residue theorem, allow computation of otherwise intractable integrals, prove that holomorphic functions are infinitely differentiable (indeed, analytic), and establish fundamental properties like the maximum modulus principle. The integral formula shows that complex differentiability is an extraordinarily strong condition—much stronger than real differentiability.

Historically, Augustin-Louis Cauchy developed these ideas in the 1820s-1840s, though his original arguments lacked the rigor of modern standards. The theorem initially appeared with restrictive hypotheses, and it took decades of work by Riemann, Weierstrass, Goursat, and others to achieve the elegant modern formulations. Goursat's 1900 proof, which eliminated the need to assume continuity of the derivative, was particularly significant.

If a function is holomorphic on and inside a closed curve, then integrating it around that curve yields zero. Moreover, the value of the function at any interior point is completely determined by its values on the boundary through an explicit integral formula—the function is "rigid" and cannot be modified locally without global consequences.

More generally, for all $n\ge 0$,

Core Theory

Main Result

We first prove this for a triangle $\Delta$, then extend to general curves.

Let $T$ be a closed triangular path in $U$. We must show ${\int }_{T}f(z)dz=0$.

Subdivide $T$ into four congruent triangular subpaths $T_1^{(1)},T_2^{(1)},T_3^{(1)},T_4^{(1)}$ by connecting midpoints. By the orientation convention, the integrals over interior edges cancel, so

At least one subtriangle satisfies $\left|{\int }_{T_j^{(1)}}f(z)dz\right|\ge \frac{1}{4}\left|{\int }_{T}f(z)dz\right|$. Call this $T^{(1)}$.

Iterate this process: construct a sequence of triangular paths $T\supseteq T^{(1)}\supseteq T^{(2)}\supseteq \cdots$ where each $T^{(n+1)}$ is a sub-triangle of $T^{(n)}$ with side length $2^{-n}$ times that of $T$, and

By the nested interval theorem, there exists a unique point $a\in {\bigcap }_{n=1}^{\infty }{T}^{(n)}$.

Since $f$ is holomorphic at $a$, we have $f(z)=f(a)+f^{\prime }(a)(z-a)+\psi (z)(z-a)$ where ${\lim }_{z\to a}\psi (z)=0$.

For the constant and linear terms, ${\int }_{T^{(n)}}[f(a)+f^{\prime }(a)(z-a)]dz=0$ (integration of polynomials around closed curves).

Thus ${\int }_{T^{(n)}}f(z)dz={\int }_{T^{(n)}}\psi (z)(z-a)dz$.

Let $L_n$ be the perimeter of $T^{(n)}$. Then $L_n=2^{-n}L$ where $L$ is the perimeter of $T$.

For $ϵ>0$, choose $\delta >0$ such that $|z-a|<\delta ⟹|\psi (z)|<ϵ$.

For $n$ large enough that the diameter of $T^{(n)}$ is less than $\delta$, we have for all $z$ on $T^{(n)}$: $|\psi (z)|<ϵ$ and $|z-a|\le L_n$.

Thus:

But also $\left|{\int }_{T^{(n)}}f(z)dz\right|\ge 4^{-n}\left|{\int }_{T}f(z)dz\right|$.

Therefore $\left|{\int }_{T}f(z)dz\right|\le ϵL^2$ for all $ϵ>0$, implying ${\int }_{T}f(z)dz=0$.

Since $U$ is simply connected, any closed curve $\gamma$ in $U$ bounds a region contained in $U$. By Green's theorem considerations or triangulation arguments, we can approximate the integral over $\gamma$ by sums of integrals over triangular paths, each of which vanishes by Step 1. Taking limits establishes the result for general piecewise-$C^1$ curves.

(A more rigorous approach uses that in a simply connected domain, $f$ has a primitive function $F$ with $F^{\prime }=f$, constructed via path integration. Then ${\oint }_{\gamma }f(z)dz=F(\gamma (b))-F(\gamma (a))=0$ since $\gamma$ is closed.) □

We may assume $n(\gamma ,a)=1$ by linearity (the general case follows by considering connected components).

As $ϵ\to 0$, by continuity of $f$, this approaches $i{\int }_0^{2\pi }f(a)d\theta =2\pi if(a)$.

But the integral is independent of $ϵ$ by Step 2, so

We prove this by induction. The case $n=0$ is Theorem 2.

Assume the formula holds for $n$. We must show:

Consider the difference quotient:

Algebraically,

Using the identity $A^{n+1}-B^{n+1}=(A-B)(A^n+A^{n-1}B+\cdots +B^n)$ with $A=z-a$ and $B=z-a-h$:

As $h\to 0$, this sum has $n+1$ terms each approaching $(z-a{)}^n$, so the ratio approaches $\frac{(n+1)}{(z-a{)}^{n+2}}$.

By the dominated convergence theorem for path integrals (justified by uniform bounds on compact sets away from $a$),

Examples and Applications

1

Evaluate ${\oint }_{|z|=2}\frac{e^z}{z^2-1}dz$.

Use partial fractions:

By Cauchy's integral formula, with $f(z)=e^z$ (holomorphic everywhere):

Therefore:

2

Find $f^{\prime \prime }(0)$ where $f(z)=\frac{1}{1-z}$ using Cauchy's formula with the contour $|z|=1/2$.

Alternatively, we can compute directly. Since $f(z)={\sum }_{n=0}^{\infty }{z}^n$ for $|z|<1$ (geometric series), we have:

Thus $f^{\prime \prime }(0)=2\cdot 1=2$.

We can verify this with the integral formula:

By the Cauchy residue theorem or direct calculation, ${\oint }_{|z|=1/2}{z}^{n-3}dz=0$ unless $n-3=-1$, i.e., $n=2$, in which case the integral is $2\pi i$.

Therefore the integral equals $2\pi i$, confirming $f^{\prime \prime }(0)=\frac{1}{\pi i}\cdot 2\pi i=2$.

Consider $f(z)=\frac{1}{z}$ and the contour $\gamma (t)=e^{2\pi it}$ for $t\in [0,1]$ (the unit circle).

The function $f$ is holomorphic on $\mathbb{C}\setminus \{0\}$, and $\gamma$ lies in this domain. However:

This does not contradict Cauchy's theorem because $\gamma$ is not null-homotopic in $\mathbb{C}\setminus \{0\}$—it winds once around the hole at the origin. The domain $\mathbb{C}\setminus \{0\}$ is not simply connected.

If we take a contour like $|z|=2$ and $f(z)=z^2$ (holomorphic everywhere), then Cauchy's theorem correctly gives ${\oint }_{|z|=2}{z}^{2}dz=0$.

The lesson: the topology of the domain relative to the contour is essential.

Cauchy's integral theorem says that if you integrate a holomorphic function around a closed loop that doesn't "trap" any singularities, you get zero. Think of the function values as defining a vector field on the plane; holomorphy imposes such rigid constraints that the circulation around any contractible loop vanishes.

The integral formula has a beautiful interpretation: the value of $f$ at a point $a$ is a weighted average of its values on a surrounding circle, weighted by the "kernel" $\frac{1}{2\pi i(z-a)}$. This is the mean value property in complex analysis. The function cannot have local extrema in the interior (leading to the maximum modulus principle) because its value is always an average of nearby values.

Physically, if $f$ represents a complex potential (e.g., in fluid dynamics or electrostatics), the integral theorem corresponds to conservation laws, and the integral formula shows that the potential at any point is determined by boundary data—a fundamental principle in potential theory.

The increasing powers in the derivative formula show that higher derivatives are increasingly sensitive to the values of $f$ near $a$, with the weighting kernel $(z-a{)}^{-(n+1)}$ giving more weight to points closer to $a$.

Historical Development

Augustin-Louis Cauchy first stated his integral theorem in 1825, though his proof relied on the assumption that $f^{\prime }$ was continuous. This was a significant restriction that Cauchy himself was aware of, and the theorem appeared in various forms throughout his work in the 1820s-1840s. Cauchy's integral formula appeared in his 1831 work.

The crucial breakthrough came with Édouard Goursat's 1900 proof, which eliminated the need to assume continuity of the derivative, showing that complex differentiability alone (the existence of $f^{\prime }(a)$ at each point) was sufficient. This "Goursat's theorem" showed that holomorphic functions are automatically continuously differentiable, a remarkable rigidity property.

The modern formulation in terms of homotopy and winding numbers emerged in the early 20th century with the development of algebraic topology. Henri Cartan and others in the 1950s generalized these ideas to several complex variables, where the situation becomes dramatically more complicated.

Connections to Other Areas

What You Need First

What This Opens Up

See Also