Cauchy's Theorem

Theorem 5.1 (Cauchy”s Theorem). If $f$ is analytic on a connected domain $D$ and $\gamma$ Is a simple closed contour in $D$Then

$\int_\gamma f(z)\, dz = 0$

Proof (for $f'$ continuous). By Green’s theorem in the plane, writing $f = u + iv$:

$\int_\gamma f\, dz = \int_\gamma (u\, dx - v\, dy) + i\int_\gamma (v\, dx + u\, dy)$

Applying Green’s theorem to each integral:

$= \iint_D (-v_x - u_y)\, dA + i\iint_D (u_x - v_y)\, dA = 0$

By the Cauchy-Riemann equations. $\blacksquare$

A domain $D \subseteq \mathbb{C}$ is ** connected** if every simple closed contour in $D$ can Be continuously shrunk to a point within $D$.

Cauchy’s theorem may fail on multiply connected domains. For example, $\int_\gamma \frac{1}{z}\, dz = 2\pi i$ where $\gamma$ is the unit circle (traversing a region that Excludes the singularity at $z = 0$).

Corollary 5.2. If $f$ is analytic on a connected domain $D$Then the integral $\int_{z_0}^{z_1} f(z)\, dz$ is independent of the path from $z_0$ to $z_1$ in $D$.

Theorem 5.3. If $f$ is analytic on a connected domain $D$Then $f$ has an antiderivative $F$ in $D$ (i.e., $F'(z) = f(z)$), and

$\int_\gamma f(z)\, dz = F(z_1) - F(z_0)$

Where $z_0$ and $z_1$ are the endpoints of $\gamma$.

Theorem 5.4. If $f$ is analytic on a domain $D$ containing simple closed contours $\gamma, \gamma_1, \ldots, \gamma_n$ where $\gamma_1, \ldots, \gamma_n$ Lie in the interior of $\gamma$ and the region between $\gamma$ and the $\gamma_k$ is contained in $D$ And all contours are positively oriented, then

$\int_\gamma f(z)\, dz = \sum_{k=1}^n \int_{\gamma_k} f(z)\, dz$

Theorem 5.5 (Deformation of Contours). If $f$ is analytic on a domain containing two simple Closed contours $\gamma_1$ and $\gamma_2$ where one can be continuously deformed into the other Within the domain of analyticity of $f$Then

$\int_{\gamma_1} f(z)\, dz = \int_{\gamma_2} f(z)\, dz$

Proof. This follows directly from Theorem 5.4 applied to the region between $\gamma_1$ and $\gamma_2$. $\blacksquare$

Remark. This theorem is enormously useful: we can replace a complicated contour with a simpler one (a small circle around each singularity) without changing the value of the integral.

• Assuming Cauchy’s theorem applies to any closed contour: The theorem requires $f$ to be analytic on a simply connected domain containing the contour. If the contour encloses any singularity, the integral may be non-zero.
• Confusing simply connected with connected: A domain can be connected but not simply connected (e.g., an annulus). Cauchy’s theorem fails on such domains without additional conditions on the contour.
• Applying the deformation theorem outside the domain of analyticity: The contour can only be deformed through regions where $f$ remains analytic. Deforming a contour across a singularity changes the value of the integral.
• Forgetting orientation when using the multiply connected theorem: The outer contour and inner contours must be traversed with consistent positive orientation (counterclockwise for the outer, clockwise for the inner) for the equality $\int_\gamma f = \sum \int_{\gamma_k} f$ to hold.

Problem. Evaluate $I = \int_0^{2\pi} \frac{d\theta}{2 + \cos\theta}$.

Solution. Let $z = e^{i\theta}$, so $d\theta = dz/(iz)$ and $\cos\theta = (z + z^{-1})/2$.

$I = \oint_{|z|=1} \frac{1}{2 + (z + z^{-1})/2} \cdot \frac{dz}{iz} = \oint_{|z|=1} \frac{2}{4z + z^2 + 1} \cdot \frac{dz}{i} = \frac{2}{i} \oint_{|z|=1} \frac{dz}{z^2 + 4z + 1}$

The denominator factors as $(z + 2 - \sqrt{3})(z + 2 + \sqrt{3})$. Only the root $z = -2 + \sqrt{3}$ lies inside $|z| = 1$. By Cauchy’s theorem applied to the simply connected region after deformation:

$I = \frac{2}{i} \cdot 2\pi i \cdot \operatorname{Res}_{z=-2+\sqrt{3}} \frac{1}{z^2 + 4z + 1} = 4\pi \cdot \frac{1}{2\sqrt{3}} = \frac{2\pi}{\sqrt{3}}$

Problem. Evaluate $\int_0^\infty \frac{\sqrt{x}}{x^2 + 1}\,dx$.

Solution. Consider $f(z) = \frac{\sqrt{z}}{z^2 + 1}$ with a branch cut along the positive real axis. Integrate around a keyhole contour $\gamma$ consisting of $C_R$ (large circle radius $R$), $C_\varepsilon$ (small circle radius $\varepsilon$), and two straight segments just above and below the cut. On the upper segment, $\sqrt{z} = \sqrt{x}$; on the lower segment, $\sqrt{z} = -\sqrt{x}$ (due to the $2\pi$ phase change). By Cauchy’s theorem:

$\int_\gamma f(z)\,dz = 2\pi i \left(\operatorname{Res}_{z=i} f(z) + \operatorname{Res}_{z=-i} f(z)\right)$

As $R \to \infty$ and $\varepsilon \to 0$, the circular contributions vanish, leaving:

$2\int_0^\infty \frac{\sqrt{x}}{x^2 + 1}\,dx = 2\pi i \left(\frac{\sqrt{i}}{2i} + \frac{\sqrt{-i}}{-2i}\right) = \frac{\pi}{\sqrt{2}}$

Hence $\int_0^\infty \frac{\sqrt{x}}{x^2 + 1}\,dx = \frac{\pi}{\sqrt{2}}$.

• Cauchy’s theorem requires analyticity on the entire region enclosed by the contour. If $f$ has even a single singularity inside $\gamma$, the integral may be nonzero.
• The integral over a closed contour equals $2\pi i$ times the sum of residues (a consequence of Cauchy’s theorem for multiply connected domains), connecting Cauchy’s theorem to the residue calculus.
• Path independence is equivalent to the existence of an antiderivative on a simply connected domain, which in turn is guaranteed by Cauchy’s theorem.
• Deformation of contours allows replacing complicated paths with simple ones (e.g., small circles around singularities) without changing the integral value.
• The Cauchy-Riemann equations are both necessary and sufficient for the proof: the vanishing of the double integral in the proof relies entirely on $u_x = v_y$ and $u_y = -v_x$.

• Evaluating real integrals: Many difficult real integrals (e.g., $\int_0^\infty \frac{\cos x}{x^2+1}\,dx$) are computed by choosing appropriate contours and applying Cauchy’s theorem.
• Computing residues: The residue theorem, which follows from Cauchy’s theorem, is the standard tool for evaluating contour integrals in physics and engineering.
• Conformal mapping: Cauchy’s theorem underpins the theory of conformal maps, used in fluid dynamics and electrostatics to solve boundary value problems.
• Signal processing: The Laplace and Fourier transforms rely on contour integration techniques derived from Cauchy’s theorem.
• Number theory: Contour integrals related to the Riemann zeta function use Cauchy’s theorem to establish properties of prime number distribution.