Holomorphic Functions and Cauchy Theory

Holomorphy is unusually rigid: values of a holomorphic function on one closed contour determine every derivative at each enclosed point. The mechanism is Cauchy’s integral formula. Its force comes with precise hypotheses—the function must be holomorphic throughout the relevant region, the evaluation point must not lie on the contour, and the contour’s orientation and winding must be retained. A contour may be deformed only through a region where those hypotheses remain true.

This page develops the finite-contour theorems and one regulated energy-plane example. Laurent expansions, systematic residue calculus, branch cuts, and pinch obstructions are treated on the later pages of this chapter.

Holomorphic functions and contour integrals

Let $\Omega\subseteq\mathbb C$ be open. A function $f:\Omega\to\mathbb C$ is holomorphic when the complex derivative

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

exists at every $z_0\in\Omega$. The increment $h$ may approach zero from any complex direction. That single requirement is much stronger than ordinary real differentiability in two variables.

A piecewise-$C^1$ path is a map $\gamma:[a,b]\to\Omega$. Its contour integral is

$$\int_\gamma f(z)\,\mathrm dz = \int_a^b f(\gamma(t))\gamma'(t)\,\mathrm dt.$$

Reparametrizing without reversing direction leaves the integral unchanged. Reversing the path changes its sign. When $\gamma(a)=\gamma(b)$, write $\oint_\gamma$.

For a closed path avoiding $w$, its winding number about $w$ is

$$\operatorname{Ind}_\gamma(w) = \frac{1}{2\pi i} \oint_\gamma\frac{\mathrm dz}{z-w}.$$

For the positively oriented circle around $w$ this number is $1$; traversing the circle clockwise gives $-1$. More complicated closed paths can have any integer winding number. Winding is constant as $w$ moves within a connected component of $\mathbb C\setminus\gamma$.

Cauchy’s theorem: the hypotheses behind zero

A practical form of Cauchy’s integral theorem is:

Let $C$ be a positively oriented, piecewise-$C^1$ Jordan curve. If $f$ is holomorphic on an open set containing $C$ and its interior, then $\oint_C f(z)\,\mathrm dz=0$.

The phrase “and its interior” is essential. Holomorphy merely on the trace of the curve is not enough. A convenient stronger setup is to place the curve in a simply connected domain $\Omega$ and require $f$ to be holomorphic throughout $\Omega$.

The corresponding general statement uses winding rather than the drawing of an “inside.” If a closed piecewise-$C^1$ cycle $\gamma$ in $\Omega$ has zero winding about every point outside $\Omega$—equivalently, it is null-homologous in $\Omega$—then

$$\oint_\gamma f(z)\,\mathrm dz=0$$

for every holomorphic $f$ on $\Omega$. This version handles self-intersections and several boundary components without silently filling holes. Conway 1978, Chapter IV gives the structural theorem and its homological form; Orloff 2018, Topic 3, PDF give the simply connected and cut-domain constructions.

Proof status. For a sufficiently smooth Jordan boundary, writing $f=u+iv$ and applying Green’s theorem to the real and imaginary parts reduces the integral to the Cauchy–Riemann equations. That argument is pedagogically useful but assumes more boundary regularity than the theorem needs. The general result follows by local holomorphic primitives and subdivision, or by the homological theorem cited above; it is stated here rather than proved in full.

Cauchy’s formula: boundary data recover local derivatives

Under the general hypotheses just stated, let $a\in\Omega\setminus\gamma$. Then for every integer $n\geq 0$,

$$\operatorname{Ind}_\gamma(a)f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\,\mathrm dz.$$

For one positive winding, the case $n=0$ is the familiar formula

$$f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm dz.$$

This is the central answer to the page’s question. The derivative $f^{(n)}(a)$ is local data, yet it is recovered from values of $f$ all along a surrounding contour. Conversely, fixing the contour values fixes every Taylor coefficient in its interior. The derivative formula and elementary examples are developed in Orloff 2018, Topic 4, PDF; the winding-number hypotheses are those of Conway, Chapter IV.

For $n=0$, the proof idea is to subtract the apparent singularity:

$$g(z) = \begin{cases} \dfrac{f(z)-f(a)}{z-a}, & z\neq a,\\[6pt] f'(a), & z=a. \end{cases}$$

The point $a$ is removable for $g$, so Cauchy’s theorem gives $\oint_\gamma g(z)\,\mathrm dz=0$. Therefore

$$\oint_\gamma\frac{f(z)}{z-a}\,\mathrm dz = f(a)\oint_\gamma\frac{\mathrm dz}{z-a} = 2\pi i\,\operatorname{Ind}_\gamma(a)f(a).$$

Higher derivatives follow by differentiating the integral representation with respect to $a$; the contour stays a positive distance away, so the differentiation is uniform on compact subsets of the interior.

Two immediate consequences make the rigidity quantitative. Suppose the closed disk $\overline{D(a,r)}$ is contained in $\Omega$, and set

$$M_r=\max_{|z-a|=r}|f(z)|,$$

then the contour-length estimate gives Cauchy’s inequalities,

$$|f^{(n)}(a)| \leq \frac{n!M_r}{r^n}.$$

Also, for $|w-a|<r$, expanding $1/(z-w)$ as a uniformly convergent geometric series on the circle yields

$$f(w) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(w-a)^n.$$

Thus a holomorphic function is analytic: it is represented locally by its Taylor series. On a connected domain, this leads to the identity theorem—two holomorphic functions that agree on a subset having an accumulation point in that domain agree everywhere there. Continuing beyond one disk still requires an overlapping chain that avoids singularities; later pages make that path dependence explicit.

Primitives, path independence, and holes

For a continuous function $f$ on a connected domain, the following statements are equivalent:

1. $\int_\gamma f(z)\,\mathrm dz$ depends only on the endpoints of $\gamma$.
2. $\oint_\gamma f(z)\,\mathrm dz=0$ for every closed path $\gamma$.
3. There is a primitive $F$ on the domain with $F'=f$.

If the domain is simply connected and $f$ is holomorphic, Cauchy’s theorem supplies these conditions. The topology cannot be omitted. The function $f(z)=1/z$ is holomorphic on $\mathbb C\setminus\{0\}$, but

$$\oint_{|z|=1}\frac{\mathrm dz}{z}=2\pi i,$$

so it has no single-valued primitive on the punctured plane.

Boundary orientation is easiest to remember by walking with the domain on the left. For the annulus

$$A=\{z:r<|z|<R\},$$

the outer boundary is counterclockwise and the inner boundary is clockwise. If $f$ is holomorphic on a neighborhood of the closed annulus, then

$$\oint_{|z|=R}^{\mathrm{CCW}} f(z)\,\mathrm dz + \oint_{|z|=r}^{\mathrm{CW}} f(z)\,\mathrm dz =0.$$

Equivalently, the two counterclockwise integrals are equal. For $f(z)=1/z$, each counterclockwise integral is $2\pi i$, while the correctly oriented total boundary integral vanishes. A hole changes which individual curves are null-homologous; it does not invalidate Cauchy’s theorem.

Smallest winding-number check

Let $C_R$ be the counterclockwise circle $|z|=R$. Direct parametrization, or Cauchy’s formula with $f=1$, gives

$$\oint_{C_R}\frac{\mathrm dz}{z-a} = \begin{cases} 2\pi i, & |a|<R,\\ 0, & |a|>R. \end{cases}$$

If the orientation is reversed, both answers change sign, so the nonzero answer becomes $-2\pi i$. If $|a|=R$, the integrand is singular on the contour and none of these formulas applies. This three-way check—inside, outside, or on the contour—should precede any contour manipulation.

QFT-facing example: a regulated energy contour

Adopt the site’s inverse-Fourier sign and let $E>0$, $\eta>0$. Define

$$E_\eta=\sqrt{E^2-i\eta}, \qquad \operatorname{Re}E_\eta>0, \qquad \operatorname{Im}E_\eta<0.$$

This fixes the square-root branch continuously from positive $E$. The standard regulated free energy denominator then factors exactly:

$$I_\eta(t) = \int_{-\infty}^{\infty} \frac{\mathrm dp^0}{2\pi}\, \frac{i\,e^{-ip^0t}} {(p^0)^2-E^2+i\eta} = \int_{-\infty}^{\infty} \frac{\mathrm dp^0}{2\pi}\, \frac{i\,e^{-ip^0t}} {(p^0-E_\eta)(p^0+E_\eta)}.$$

Its poles are

$$z_+=E_\eta, \qquad z_-=-E_\eta.$$

For a finite positively oriented contour $C$ enclosing $z_+$ but not $z_-$, Cauchy’s formula applies to the holomorphic numerator

$$h(z)=\frac{i\,e^{-izt}}{z-z_-}$$

and gives

$$\oint_C \frac{i\,e^{-izt}} {(z-z_+)(z-z_-)}\,\mathrm dz = -\,\frac{2\pi e^{-iz_+t}}{z_+-z_-}.$$

That is a finite-contour statement. Turning the real integral into a closed contour requires a separate estimate. For $t>0$, the factor

$$|e^{-izt}|=e^{t\,\operatorname{Im}z}$$

decays in the lower half-plane. On a lower semicircle of radius $R$, it is at most $1$, the denominator is of order $R^2$, and the arc length is $\pi R$, so the arc contribution is $O(R^{-1})$. The lower closure is clockwise, and therefore

$$I_\eta(t) = \frac{e^{-iz_+t}}{z_+-z_-}, \qquad t>0.$$

For $t<0$, the upper semicircle is the decaying one and encloses $z_-$ with counterclockwise orientation:

$$I_\eta(t) = \frac{e^{-iz_-t}}{z_+-z_-}, \qquad t<0.$$

Consequently,

$$\lim_{\eta\to0^+}I_\eta(t) = \frac{e^{-iE|t|}}{2E}.$$

This calculation uses Cauchy’s formula for a single linear factor and an explicit large-arc bound. The next page, Laurent Series, Poles, and Residues, systematizes such local contributions. The interpretation in terms of time ordering and distinct two-point functions belongs to Scalar Propagators, Ordered Correlators, and Sources. The physical application and its pole placement are given in Schwartz 2014, § 6.2, pp. 75–77, Fig. 6.1 and Eqs. (6.28)–(6.34). Schwartz writes $e^{+i\omega\tau}$ and therefore closes upward for $\tau>0$. The site convention follows after $p^0=-\omega$ and $t=\tau$: the same pole is then in the lower $p^0$-half-plane, so the apparently opposite closure directions agree.

Failure modes and stopping conditions

Holomorphic on the curve is not holomorphic inside it. A singularity in the enclosed or swept region changes the integral. Mark the whole region, not only the initial and final contours.

A point on the contour is neither inside nor outside. Cauchy’s formula requires the kernel pole to avoid the path. Indentations, principal values, and distributional boundary values need additional definitions.

A hole cannot be filled by a sketch. In a multiply connected domain, check winding numbers or include every boundary component with its induced orientation.

A deformation and a closure are different steps. Cauchy’s theorem can compare two finite contours through a holomorphic region. Adding an arc at infinity also requires a uniform decay bound on that arc.

The $i0$ limit is not an ordinary substitution. Keep $\eta>0$ while locating poles and proving estimates. The limit can be distributional, especially when poles approach the real axis.

Exercises

1. Evaluate $\oint_{|z|=2} e^z z^{-3}\,\mathrm dz$ with counterclockwise orientation.

   Check

   Use the derivative formula with $a=0$ and $n=2$:

   $$\oint_{|z|=2}\frac{e^z}{z^3}\,\mathrm dz = \frac{2\pi i}{2!} \left.\frac{\mathrm d^2}{\mathrm dz^2}e^z\right|_{z=0} = \pi i.$$

2. Let $A=\{1<|z|<3\}$. Explain why the positively oriented boundary integral of $1/z$ is zero even though each counterclockwise circle has integral $2\pi i$.

   Check

   The positive boundary orientation is the outer circle counterclockwise and the inner circle clockwise. Their contributions are $2\pi i$ and $-2\pi i$, respectively. The sum is zero, in agreement with holomorphy of $1/z$ on the annulus.

3. For $t>0$, close $\int_{\mathbb R} e^{-izt}(z^2+1)^{-1}\,\mathrm dz$ in the lower half-plane. State the arc estimate and account for the orientation.

   Check

   On the lower arc, $|e^{-izt}|\leq1$, while $|z^2+1|$ is of order $R^2$ and the length is $\pi R$, so the arc is $O(R^{-1})$. The lower contour is clockwise and encloses $z=-i$. Applying Cauchy’s formula to the factor $z+i$ gives the real-line value $\pi e^{-t}$.

References

• John B. Conway, Functions of One Complex Variable I, 2nd ed., Chapter IV, Springer, 1978. Complex Integration chapter. This is the structural source for Cauchy’s theorem, winding numbers, the general integral formula, and exact hypotheses.
• Jeremy Orloff, 18.04 Complex Variables with Applications, MIT OpenCourseWare, 2018: Topic 3, Line Integrals and Cauchy’s Theorem, PDF and Topic 4, Cauchy’s Integral Formula, PDF. These notes supply the elementary contour constructions, examples, derivative formula, and contour-length estimates.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, §6.2, Cambridge University Press, 2014. Book record. This is the QFT source for the regulated energy integral and Feynman pole placement.