Cauchy's Integral Formula

⚡ Key Takeaways

Cauchy's Integral Formula states that the value of a holomorphic function at an interior point equals a contour integral of the function around an enclosing loop: $f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz$ . Key result in words: the values of an analytic function on a closed curve completely determine its values everywhere inside. Where it applies: complex analysis, evaluation of real integrals, signal processing, fluid dynamics, quantum field theory. Who/when: Augustin-Louis Cauchy, 1831 (with foundations laid in the 1820s).

💡 The Core Insight

A holomorphic function is so tightly constrained that its boundary values broadcast inward, determining everything in the interior. The kernel $\frac{1}{z-a}$ acts like a "sampling probe" that, when integrated around a loop, sifts out exactly the value $f(a)$ . This single mechanism is responsible for analyticity, the maximum modulus principle, Liouville's theorem, and the residue calculus.

DefinitionHolomorphic Function and Contour

A function $f:\Omega \to \mathbb{C}$ defined on an open set $\Omega \subseteq \mathbb{C}$ is holomorphic at $z_0$ if the limit

1

\quad f'(z_0) = \lim_{h\to 0}\frac{f(z_0+h)-f(z_0)}{h}

Derivation hint: the limit must exist and be independent of the direction in which the complex increment $h\to 0$ , a far stronger condition than real differentiability. exists, where $h\in\mathbb{C}$ . A contour (or closed curve) $C$ is a piecewise smooth, continuous path $\gamma:[a,b]\to\mathbb{C}$ ; it is simple closed if $\gamma(a)=\gamma(b)$ and $\gamma$ is injective on $(a,b)$ . It is positively oriented if traversed counterclockwise, keeping the enclosed region on the left.

TheoremCauchy's Integral Formula

Let $f$ be holomorphic on an open set containing the closed region bounded by a positively oriented simple closed contour $C$ . Then for any point $a$ in the interior of $C$ ,

2

\quad f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz

Derivation hint: deform $C$ to a small circle around $a$ using <span class="wiki-link-missing">Cauchy's Integral Theorem</span>, then evaluate the resulting integral as the radius shrinks. More generally, $f$ has derivatives of all orders, given by the generalized Cauchy formula:

3

\quad f^{(n)}(a) = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z-a)^{n+1}}\,dz

Derivation hint: differentiate [2] $n$ times with respect to the parameter $a$ under the integral sign, justified by uniform continuity of the kernel on compact sets away from $C$ . Here $\oint_C$ denotes the line integral around $C$ , $i=\sqrt{-1}$ , and $2\pi i$ is the value of $\oint_C \frac{dz}{z-a}$ for any simple loop enclosing $a$ once.

ProofDerivation of the Integral Formula

We prove [2] in numbered steps.

Step 1 — Set up an auxiliary function. Fix $a$ in the interior of $C$ . Define

4

\quad g(z) = \frac{f(z)-f(a)}{z-a}, \qquad z\neq a

Derivation hint: subtract the constant $f(a)$ so that the singularity at $z=a$ becomes removable, exploiting differentiability of $f$ there. Since $f$ is holomorphic, $g$ has a removable singularity at $a$ because $\lim_{z\to a} g(z) = f'(a)$ exists and is finite; thus $g$ extends to a holomorphic function on the whole region.

Step 2 — Apply Cauchy's Integral Theorem to $g$ . Because $g$ is holomorphic inside and on $C$ , Cauchy's Integral Theorem gives

5

\quad \oint_C g(z)\,dz = 0 \;\Longrightarrow\; \oint_C \frac{f(z)}{z-a}\,dz = f(a)\oint_C \frac{dz}{z-a}

Derivation hint: split the integrand $g(z)$ into $\frac{f(z)}{z-a}$ minus $\frac{f(a)}{z-a}$ and use that the integral of the holomorphic $g$ vanishes.

Step 3 — Evaluate the kernel integral. It remains to compute $\oint_C \frac{dz}{z-a}$ . By contour deformation, replace $C$ with a small circle $C_\varepsilon$ of radius $\varepsilon$ centered at $a$ , parametrized $z = a + \varepsilon e^{i\theta}$ , $dz = i\varepsilon e^{i\theta}\,d\theta$ :

6

\quad \oint_{C_\varepsilon}\frac{dz}{z-a} = \int_0^{2\pi}\frac{i\varepsilon e^{i\theta}}{\varepsilon e^{i\theta}}\,d\theta = \int_0^{2\pi} i\,d\theta = 2\pi i

Derivation hint: the radius cancels in the integrand; the winding-number factor $2\pi i$ is intrinsic to a single counterclockwise loop.

Step 4 — Combine. Substituting [6] into [5] yields $\oint_C \frac{f(z)}{z-a}\,dz = 2\pi i\, f(a)$ , which is precisely [2] after dividing by $2\pi i$ . $\blacksquare$

∎

ProofSketch for the Derivative Formula

To obtain [3], write the difference quotient of [2]:

7

\quad \frac{f(a+h)-f(a)}{h} = \frac{1}{2\pi i}\oint_C f(z)\left[\frac{1}{h}\left(\frac{1}{z-a-h}-\frac{1}{z-a}\right)\right]dz

Derivation hint: combine the two fractions over a common denominator; the bracket simplifies to $\frac{1}{(z-a-h)(z-a)}$ . As $h\to 0$ the bracket tends to $\frac{1}{(z-a)^2}$ uniformly on $C$ (which is bounded away from $a$ ), so passage to the limit under the integral gives $f'(a)=\frac{1}{2\pi i}\oint_C \frac{f(z)}{(z-a)^2}\,dz$ . Induction on $n$ , repeating this argument, produces the general formula [3]. $\blacksquare$

∎

mathematician · Augustin-Louis Cauchy | 1789–1857

Cauchy was a prolific French mathematician who, almost single-handedly, made analysis rigorous. Working in Paris during a turbulent era of revolution and restoration, he introduced precise definitions of limits, continuity, and convergence, and founded the modern theory of functions of a complex variable. His 1825 memoir on definite integrals between complex limits and his 1831 work in Turin gave the integral theorem and integral formula that bear his name. Famously devout and politically royalist, Cauchy refused oaths of allegiance and went into self-exile, yet never stopped publishing — producing nearly 800 papers across analysis, elasticity, optics, and celestial mechanics.

🧠 Intuition

Picture the kernel $\frac{1}{z-a}$ as a probe that "tunes in" to the point $a$ . As $z$ travels once around the loop $C$ , the factor $\frac{1}{z-a}$ winds once around the origin, contributing the universal phase factor $2\pi i$ . Multiplying by $f(z)$ and integrating averages the boundary values, but the winding singularity at $a$ pulls the average toward $f(a)$ exactly. Another picture: holomorphic functions behave like incompressible, irrotational flows; their boundary behavior rigidly determines the interior, just as the boundary velocity of an ideal fluid determines the entire flow pattern. This is why analytic continuation works — once you know $f$ on any small piece, the formula forces its values everywhere accessible.

Figure

Contour C enclosing the point a, with deformation to a small circle

ExampleEvaluating a Contour Integral Directly

Given: Compute $\displaystyle\oint_C \frac{e^{z}}{z-2}\,dz$ where $C$ is the circle $|z|=3$ traversed counterclockwise.

Steps: 1. Identify $f(z)=e^z$ and the singularity $a=2$ . 2. Check that $a=2$ lies inside $C$ : since $|2|=2<3$ , yes. 3. Confirm $f(z)=e^z$ is holomorphic everywhere (entire), so the hypotheses of [2] hold. 4. Apply Cauchy's Integral Formula [2] rearranged: $\oint_C \frac{f(z)}{z-a}\,dz = 2\pi i\, f(a)$ . 5. Compute $f(2)=e^2$ .

Answer:

8

\quad \oint_C \frac{e^{z}}{z-2}\,dz = 2\pi i\, e^{2} \approx \boxed{2\pi i\,(7.389) = 46.42\,i}

Derivation hint: substitute $a=2$ and $f(a)=e^2$ into $2\pi i f(a)$ ; the result is purely imaginary because $e^2$ is real.

ExampleUsing the Derivative Formula

Given: Compute $\displaystyle\oint_C \frac{\cos z}{(z-0)^{3}}\,dz$ where $C$ is the unit circle $|z|=1$ , counterclockwise.

Steps: 1. Match to formula [3] with $f(z)=\cos z$ , $a=0$ , and exponent $n+1=3$ , so $n=2$ . 2. The formula reads $\oint_C \frac{f(z)}{(z-a)^{3}}\,dz = \frac{2\pi i}{2!}f''(a)$ . 3. Differentiate: $f'(z)=-\sin z$ , $f''(z)=-\cos z$ . 4. Evaluate $f''(0)=-\cos 0 = -1$ . 5. Insert: $\frac{2\pi i}{2}(-1)= -\pi i$ .

Answer:

9

\quad \oint_C \frac{\cos z}{z^{3}}\,dz = \frac{2\pi i}{2!}\,f''(0) = \boxed{-\pi i}

Derivation hint: the factor $\frac{n!}{2\pi i}$ in [3] inverts to $\frac{2\pi i}{n!}$ when solving for the integral; here $n=2$ gives $\frac{2\pi i}{2}=\pi i$ .

⚠️ Common mistake

The point $a$ must lie strictly inside $C$ . If $a$ is outside the contour, the integrand $\frac{f(z)}{z-a}$ is holomorphic on and inside $C$ , so by Cauchy's Integral Theorem the integral is simply zero — not $2\pi i\,f(a)$ . Also, the formula requires $f$ to be holomorphic on the entire closed region bounded by $C$ ; a single extra singularity inside breaks it. Finally, orientation matters: a clockwise contour flips the sign, giving $-2\pi i\,f(a)$ . The winding number around $a$ , not the geometric shape of $C$ , governs the multiplier.

Counterexample

Consider $f(z)=\bar z$ (complex conjugate). One might hope to apply the formula, but $f$ is not holomorphic anywhere — it fails the Cauchy–Riemann equations. Indeed $\oint_{|z|=1}\frac{\bar z}{z}\,dz$ . On $|z|=1$ we have $\bar z = 1/z$ , so the integrand is $1/z^2$ , and $\oint_{|z|=1}\frac{dz}{z^2}=0$ , which does not equal $2\pi i\,\bar a = 2\pi i\cdot 0 = 0$ by coincidence here — but for general $a$ inside, $f(a)=\bar a$ would be required, which the integral cannot reproduce. The formula simply does not apply to non-holomorphic functions; matching values is accidental and breaks down immediately for other contours or points.

connection

Cauchy's Integral Formula sits at the hub of complex analysis. It is an immediate consequence of Cauchy's Integral Theorem (the vanishing of integrals of holomorphic functions over closed loops). Its derivative form [3] proves that holomorphic functions are infinitely differentiable, hence equal to their Taylor Series — the principle of Analyticity of Holomorphic Functions. Extracting the coefficient of $(z-a)^{-1}$ generalizes the formula into the Residue Theorem, the master tool for evaluating real integrals. It directly yields Liouville's Theorem (bounded entire functions are constant), the Maximum Modulus Principle, and the Fundamental Theorem of Algebra. The mean value form (taking $C$ a circle) connects to Harmonic Functions and the Poisson integral.

🔬 Where this appears

⏳ History

- 1814 — Cauchy presents an early memoir on definite integrals, planting the seeds of complex integration while studying real integrals via complex methods.

- 1822–1825 — Cauchy develops the integral theorem (closed-contour integrals of holomorphic functions vanish), published in his 1825 Mémoire sur les intégrales définies prises entre des limites imaginaires.

- 1831 — While in exile in Turin, Cauchy states the integral formula and his "calcul des limites" (later the Cauchy estimates), giving the power-series representation of analytic functions.

- 1843–1846 — Cauchy systematizes the residue theorem, generalizing the integral formula to functions with poles.

- 1851 — Bernhard Riemann's dissertation recasts complex functions geometrically via the Cauchy–Riemann equations and Riemann surfaces, deepening the conceptual foundation.

- 1860s — Karl Weierstrass develops the rigorous power-series ("analytic element") approach, providing an alternative foundation independent of integration.

- 1883 — Édouard Goursat proves the integral theorem assuming only differentiability (not continuous derivatives), removing a hidden hypothesis and yielding the modern Cauchy–Goursat theorem.

- 1900s — The homological/winding-number formulation (via the index of a curve) generalizes the formula to arbitrary cycles in multiply connected regions.

open-problem

The classical formula assumes $f$ holomorphic on a simply connected region. Subtleties arise in multiply connected domains, where the general statement uses the winding number $n(C,a)=\frac{1}{2\pi i}\oint_C \frac{dz}{z-a}$ , giving $n(C,a)\,f(a)=\frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz$ — the loop may encircle $a$ several times or not at all. Extending integral-representation ideas to several complex variables leads to the Bochner–Martinelli and Cauchy–Fantappiè kernels, where the geometry is vastly richer and many natural questions (e.g., explicit reproducing kernels on general pseudoconvex domains, behavior of the $\bar\partial$ -Neumann problem) remain active research. In a different direction, Cauchy-type integrals with rough boundary data connect to deep open questions in harmonic analysis (the Cauchy transform on non-smooth curves, related to the Calderón program), some of which were only resolved late in the 20th century and whose higher-dimensional analogues persist.

Formal Statement

Why It Holds

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also