Cauchy's Theorem and Its Applications

Theorem 377.1(Goursat).

If $\Omega$ is an open set in $\mathbb{C}$, and $T\subseteq \Omega$ a triangle whose interior is also contained in $\Omega$, then

$${\int }_{T}f(z)dz=0$$

whenever $f$ is holomorphic in $\Omega$.

Proof.

Let $T\subseteq \Omega$. Subdivide $T$ into 4 congruent triangles. Since integrating the same curve in opposite directions yields zero, we can express the integral of $f$ over $T$ as the sum of the integral of $f$ over the four triangles. Continue the process to obtain a point $z_0$ in the intersection. Use the holomorphicity of $f$ at $z_0$ and bound the error term by ${ϵ}_n{d}^{(n)}{p}^{(n)}$.□

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Corollary 377.2(Corollary).

If $f$ is holomorphic in an open set $\Omega$ that contains a rectangle $R$ and its interior, then

$${\int }_{R}f(z)dz=0.$$

Local existence of primitives and Cauchy’s theorem in a disc

Lemma 377.3(Stein & Shakarchi (2003) 2.2.1).

A holomorphic function in an open disc has a primitive in that disc.

Proof.

Define $F(z)$ as the integral over a unique rectangular path from the origin to $z$.

To show that $F$ is holomorphic in $D$ and $F^{\prime }(z)=f(z)$, use Thm 1 to write

$$F(z+h)-F(z)={\int }_{\eta }f(w)dw,$$

where $\eta$ is the straight line segment from $z$ to $z+h$. Now use the continuity of $f$ to write $f(w)=f(z)+\psi (w)$, and obtain

$$\underset{h\to 0}{\lim }\frac{F(z+h)-F(z)}{h}=f(z).$$ □

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The proof of Lem 3 also works when $f$ is given to be continuous in an open disk and its integral on any triangle contained in that disk is zero; we use this observation in the proof of Thm 14.

Theorem 377.4(Cauchy's theorem for a disc).

If $f$ is holomorphic in a disc, then

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

for any closed curve $\gamma$ in that disc.

Proof.

Follows from Lem 3 and Cor 370.13.□

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Toy contours

We call a toy contour any closed curve where the notion of interior is “obvious”, and a construction similar to that in Lem 3 is possible in a neighborhood of the curve and its interior. These are useful in evaluating integrals.

To be precise (ish), let if $\Gamma$ be a toy contour and $f$ be holomorphic in a neighborhood of $\Gamma$ and its interior. Then, $f$ is holomorphic inside of a slightly larger version of $\Gamma$ whose interior contains $\Gamma$ and ${\Gamma }^{\circ }$. We fix a point $z_0\in {\Gamma }^{\circ }$. For $z\in {\Gamma }^{\circ }$, let ${\gamma }_z$ denote any curve contained inside ${\Gamma }^{\circ }$ connecting $z_0$ to $z$ which consists of finitely many horizontal and vertical segments; the choice doesn’t matter since the integral of $f$ over any two such curves would be equal, courtesy Thm 1. We may thus define $F$ unambiguously in ${\Gamma }^{\circ }$.

Thus, for a toy contour $\Gamma$, we have

$${\int }_{\Gamma }f(z)dz=0$$

whenever $f$ is holomorphic in an open set that contains the contour $\gamma$ and its interior.

Cauchy’s integral formulas

Theorem 377.5(Stein & Shakarchi (2003) 2.4.1).

Suppose $f$ is holomorphic in an open set that contains the closure of a disc $D$. If $C$ denotes the boundary circle of this disc with the positive orientation, then for any $z\in D$,

$$f(z)=\frac{1}{2\pi i}{\int }_C\frac{f(\zeta )}{\zeta -z}d\zeta .$$

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Integrate over the keyhole contour.

The regularity of holomorphic functions arises as a corollary.

Corollary 377.6(Corollary).

Let $f$ be holomorphic in an open set $\Omega$. If $C\subseteq \Omega$ is a circle whose interior is also contained in $\Omega$, then for all $z$ in the interior of $C$,

$$f^{(n)}(z)=\frac{n!}{2\pi i}{\int }_C\frac{f(\zeta )}{(\zeta -z{)}^{n+1}}d\zeta .$$

Thus, if $f$ is holomorphic in an open set $\Omega$, then $f$ has infinitely many complex derivatives in $\Omega$.

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Proof by induction.

The formulas of Thm 5 and Cor 6 are called the Cauchy integral formulas.

Corollary 377.7(Cauchy inequalities).

If $f$ is holomorphic in an open set that contains the closure of a disc $D$ centered at $z_0$ and of radius $R$, then

$$|f^{(n)}(z_0)|⩽\frac{n!\Vert f{\Vert }_C}{R^n},$$

where $\Vert f{\Vert }_C={\sup }_{z\in C}|f(z)|$.

Proof.

An easy application of Cor 6.□

We have seen (Thm 370.8) that a power series defines a holomorphic function in its disc of convergence. The converse arises as another corollary of the Cauchy integral formulas.

Theorem 377.8(Theorem).

Suppose $f$ is holomorphic in an open set $\Omega$. If $D$ is a disc centered at $z_0$ and whose closure is contained in $\Omega$, then $f$ has a power series expansion at $z_0$

$$f(z)=\sum _{n=0}^{\infty }{a}_n(z-z_0{)}^n$$

for all $z\in D$, and the coefficients are given by

$$a_n=\frac{f^{(n)}(z_0)}{n!}=\frac{1}{2\pi i}{\int }_C\frac{f(\zeta )}{(\zeta -z_0{)}^{n+1}}d\zeta$$

for all $n⩾0$.

Proof.

Use Thm 5, and write

$$\frac{1}{\zeta -z}=\frac{1}{\zeta -z_0-(z-z_0)}=\frac{1}{(\zeta -z_0)}\frac{1}{1-\frac{z-z_0}{\zeta -z_0}}.$$ □

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Since power series define infinitely differentiable functions, Thm 8 gives another proof that a holomorphic function function is infinitely differentiable.

Corollary 377.9(Corollary).

If $f$ is holomorphic on all of $\mathbb{C}$, Thm 8 implies that $f$ has a power series expansion around $0$ that converges in all of $\mathbb{C}$.

Corollary 377.10(Liouville).

If $f$ is entire and bounded, then $f$ is constant.

Proof.

It suffices to show that $f^{\prime }=0$ by Cor 370.14. For each $z_0\in \mathbb{C}$ and all $R>0$, the Cauchy inequalities yield

$$|f^{\prime }(z_0)|⩽B/R$$

where $B$ is a bound for $f$. Letting $R\to \infty$ gives the desired result.□

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Example 377.11(The fundamental theorem of algebra).

Every non-constant polynomial $P(z)=a_n{z}^n+\cdots +a_0$ with complex coefficients has a root in $\mathbb{C}$.

Proof.

If$P$ has no roots, then $1/P(z)$ is a bounded holomorphic function. To see this, assume $a_n\ne 0$ and write

$$P(z)/z^n=a_n+\left(\frac{a_{n-1}}{z}+\cdots +\frac{a_0}{z^n}\right)$$

for $z\ne 0$. Since each term in the parenthesis goes to $0$ as $|z|\to \infty$ we conclude that there exists $R>0$ such that

$$|P(z)|⩾\frac{|a_n|}{2}|z^n|$$

whenever $|z|>R$. Thus, $P$ is bounded from below when $|z|>R$. Since $P$ is continuous and has no roots in the compact disc $|z|⩽R$, it is bounded from below in that disc by Thm 138.8. This proves the claim.□

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Analytic continuation

Theorem 377.12(Theorem).

Suppose $f$ is a holomorphic in a connected region $\Omega$ that vanishes on a sequence of distinct points with a limit point in $\Omega$. Then $f$ is identically zero.

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Corollary 377.13(Corollary).

Suppose $f$ and $g$ are holomorphic in a region $\Omega$ and $f(z)=g(z)$ for all $z$ in some non-empty open subset of $\Omega$ (or more generally for $z$ in some sequence of distinct points with limit point in $\Omega$). Then $f(z)=g(z)$ throughout $\Omega$.

Morera’s theorem

The converse of Cauchy’s theorem.

Theorem 377.14(Morera).

Suppose $f$ is a continuous function in the open disc $D$ such that for any triangle $T$ contained in $D$

$${\int }_{T}f(z)dz=0,$$

then $f$ is holomorphic.

Proof.

By the proof of Lem 3, $f$ has a primitive $F$ in $D$ that satisfies $F^{\prime }=f$. By the regularity theorem, we know that $F$ is indefinitely (and hence twice) complex differentiable, and therefore $f$ is holomorphic.□

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Sequences of holomorphic functions

Theorem 377.15(Stein & Shakarchi (2003) 2.5.2-3).

If $\{f_n{\}}_{n=1}^{\infty }$ is a sequence of holomorphic functions that converges uniformly to a function $f$ in every compact subset of $\Omega$, then

1. $f$ is holomorphic in $\Omega$.
2. the sequence of derivatives $\{f_n^{\prime }{\}}_{n=1}^{\infty }$ converges uniformly to $f^{\prime }$ on every compact subset of $\Omega$.

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Compare with the real analytic version of the same theorem; Thm 15.2 goes from being a conclusion to being a hypothesis.

Holomorphic functions defined in terms of integrals

Theorem 377.16(Theorem).

Let $F(z,s)$ be defined for $(z,s)\in \Omega \times [0,1]$ where $\Omega$ is an open set in $\mathbb{C}$. Suppose $F$ satisfies the following properties:

1. $F(z,s)$ is holomorphic in $z$ for each $s$.
2. $F$ is continuous on $\Omega \times [0,1]$.

Then the function $f$ defined on $\Omega$ by

$$f(z):={\int }_0^{1}F(z,s)ds$$

is holomorphic.

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Schwarz reflection principle

Lemma 377.17(The symmetry principle).

If $f^{+}$ and $f^{-}$ are holomorphic functions in ${\Omega }^{+}$ and ${\Omega }^{-}$ respectively, that extend continuously to $I$ and $f^{+}(x)=f^{-}(x)$ for all $x\in I$, then $f$ defined on $\Omega$ by

$$f(z)=\{\begin{array}{ll}{f}^{+}(z)& z\in {\Omega }^{+}\\ f^{+}=f^{-}(z)& z\in I\\ f^{-}(z)& z\in {\Omega }^{-}\end{array}$$

is holomorphic on all of $\Omega$.

Theorem 377.18(Schwarz).

Suppose $f$ is a holomorphic function in ${\Omega }^{+}$ that extends continuously to $I$ such that $f$ is real-valued on $I$. Then there exists a function $F$ holomorphic in all of $\Omega$ such that $F=f$ on ${\Omega }^{+}$.

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References

Stein, E. M., & Shakarchi, R. (2003). Complex Analysis. Princeton University Press.