Cauchy's integral theorem

In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if $f(z)$ is holomorphic in a simply connected domain Ω, then for any simply closed contour $C$ in Ω, that contour integral is zero.

\int _{C}f(z)\,dz=0.

Statement

Fundamental theorem for complex line integrals

If $f (z)$ is a holomorphic function on an open region $U$ , and $\gamma$ is a curve in $U$ from $z_{0}$ to $z_{1}$ then,

\int _{\gamma }f'(z)\,dz=f(z_{1})-f(z_{0}).

Also, when $f (z)$ has a single-valued antiderivative in an open region $U$ , then the path integral ${\textstyle \int _{\gamma }f'(z)\,dz}$ is path independent for all paths in $U$ .

Formulation on simply connected regions

Let $U\subseteq \mathbb {C}$ be a simply connected open set, and let $f:U\to \mathbb {C}$ be a holomorphic function. Let ${\displaystyle \gamma$ be a smooth closed curve. Then:

\int _{\gamma }f(z)\,dz=0.

(The condition that $U$ be simply connected means that $U$ has no "holes", or in other words, that the fundamental group of $U$ is trivial.)

General formulation

Let $U\subseteq \mathbb {C}$ be an open set, and let $f:U\to \mathbb {C}$ be a holomorphic function. Let ${\displaystyle \gamma$ be a smooth closed curve. If $\gamma$ is homotopic to a constant curve, then:

\int _{\gamma }f(z)\,dz=0.

(Recall that a curve is homotopic to a constant curve if there exists a smooth homotopy (within $U$ ) from the curve to the constant curve. Intuitively, this means that one can shrink the curve into a point without exiting the space.) The first version is a special case of this because on a simply connected set, every closed curve is homotopic to a constant curve.

Main example

In both cases, it is important to remember that the curve $\gamma$ does not surround any "holes" in the domain, or else the theorem does not apply. A famous example is the following curve:

\gamma (t)=e^{it}\quad t\in \left[0,2\pi \right],

which traces out the unit circle. Here the following integral:

\int _{\gamma }{\frac {1}{z}}\,dz=2\pi i\neq 0,

is nonzero. The Cauchy integral theorem does not apply here since $f(z)=1/z$ is not defined at $z=0$ . Intuitively, $\gamma$ surrounds a "hole" in the domain of $f$ , so $\gamma$ cannot be shrunk to a point without exiting the space. Thus, the theorem does not apply.

Discussion

As Édouard Goursat showed, Cauchy's integral theorem can be proven assuming only that the complex derivative $f'(z)$ exists everywhere in $U$ . This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable.

The condition that $U$ be simply connected means that $U$ has no "holes" or, in homotopy terms, that the fundamental group of $U$ is trivial; for instance, every open disk $U_{z_{0}}=\{z:\left|z-z_{0}\right|<r\}$ , for $z_{0}\in \mathbb {C}$ , qualifies. The condition is crucial; consider

\gamma (t)=e^{it}\quad t\in \left[0,2\pi \right]

which traces out the unit circle, and then the path integral

\oint _{\gamma }{\frac {1}{z}}\,dz=\int _{0}^{2\pi }{\frac {1}{e^{it}}}(ie^{it}\,dt)=\int _{0}^{2\pi }i\,dt=2\pi i

is nonzero; the Cauchy integral theorem does not apply here since $f(z)=1/z$ is not defined (and is certainly not holomorphic) at $z=0$ .

One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of calculus: let $U$ be a simply connected open subset of $\mathbb {C}$ , let $f:U\to \mathbb {C}$ be a holomorphic function, and let $\gamma$ be a piecewise continuously differentiable path in $U$ with start point $a$ and end point $b$ . If $F$ is a complex antiderivative of $f$ , then

\int _{\gamma }f(z)\,dz=F(b)-F(a).

The Cauchy integral theorem is valid with a weaker hypothesis than given above, e.g. given $U$ , a simply connected open subset of $\mathbb {C}$ , we can weaken the assumptions to $f$ being holomorphic on $U$ and continuous on ${\textstyle {\overline {U}}}$ and $\gamma$ a rectifiable simple loop in ${\textstyle {\overline {U}}}$ .[1]

The Cauchy integral theorem leads to Cauchy's integral formula and the residue theorem.

Proof

If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proven as a direct consequence of Green's theorem and the fact that the real and imaginary parts of $f=u+iv$ must satisfy the Cauchy–Riemann equations in the region bounded by $\gamma$ , and moreover in the open neighborhood $U$ of this region. Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.

We can break the integrand $f$ , as well as the differential $dz$ into their real and imaginary components:

f=u+iv

dz=dx+i\,dy

In this case we have

\oint _{\gamma }f(z)\,dz=\oint _{\gamma }(u+iv)(dx+i\,dy)=\oint _{\gamma }(u\,dx-v\,dy)+i\oint _{\gamma }(v\,dx+u\,dy)

By Green's theorem, we may then replace the integrals around the closed contour $\gamma$ with an area integral throughout the domain $D$ that is enclosed by $\gamma$ as follows:

\oint _{\gamma }(u\,dx-v\,dy)=\iint _{D}\left(-{\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}\right)\,dx\,dy

\oint _{\gamma }(v\,dx+u\,dy)=\iint _{D}\left({\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}\right)\,dx\,dy

But as the real and imaginary parts of a function holomorphic in the domain $D$ , $u$ and $v$ must satisfy the Cauchy–Riemann equations there:

{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}

{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}

We therefore find that both integrands (and hence their integrals) are zero

\iint _{D}\left(-{\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}\right)\,dx\,dy=\iint _{D}\left({\frac {\partial u}{\partial y}}-{\frac {\partial u}{\partial y}}\right)\,dx\,dy=0

\iint _{D}\left({\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}\right)\,dx\,dy=\iint _{D}\left({\frac {\partial u}{\partial x}}-{\frac {\partial u}{\partial x}}\right)\,dx\,dy=0

This gives the desired result

\oint _{\gamma }f(z)\,dz=0

References

Walsh, J. L. (1933-05-01). "The Cauchy-Goursat Theorem for Rectifiable Jordan Curves". Proceedings of the National Academy of Sciences. 19 (5): 540–541. doi:10.1073/pnas.19.5.540. ISSN 0027-8424. PMC 1086062. PMID 16587781.

Kodaira, Kunihiko (2007), Complex Analysis, Cambridge Stud. Adv. Math., 107, CUP, ISBN 978-0-521-80937-5
Ahlfors, Lars (2000), Complex Analysis, McGraw-Hill series in Mathematics, McGraw-Hill, ISBN 0-07-000657-1
Lang, Serge (2003), Complex Analysis, Springer Verlag GTM, Springer Verlag
Rudin, Walter (2000), Real and Complex Analysis, McGraw-Hill series in mathematics, McGraw-Hill

External links

"Cauchy integral theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Cauchy Integral Theorem". MathWorld.
Jeremy Orloff, 18.04 Complex Variables with Applications Spring 2018 Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons.

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[1] Walsh, J. L. (1933-05-01). "The Cauchy-Goursat Theorem for Rectifiable Jordan Curves". Proceedings of the National Academy of Sciences. 19 (5): 540–541. doi:10.1073/pnas.19.5.540. ISSN 0027-8424. PMC 1086062. PMID 16587781.