# Difference between revisions of "Holomorphic function"

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− | A '''holomorphic''' | + | A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a |

+ | differentiable [[complex number|complex]] [[function]]. That is, just | ||

+ | as in the [[real number|real]] case, <math>f</math> is holomorphic at <math>z</math> if | ||

+ | <math>\lim_{h\to 0} \frac{f(z+h)-f(z)}{h}</math> exists. This is much stronger | ||

+ | than in the real case since we must allow <math>h</math> to approach zero from | ||

+ | any direction in the [[complex plane]]. | ||

+ | |||

+ | Usually, we speak of functions as holomorphic on (open) sets, | ||

+ | rather than at points, for when we consider the behavior of a function | ||

+ | at a point, we prefer to consider it in the context of the points | ||

+ | nearby. | ||

== Cauchy-Riemann Equations == | == Cauchy-Riemann Equations == | ||

− | + | We can obtain an equivalent definition if we break <math>f</math> and <math>z</math> | |

− | * < | + | into real and imaginary components. |

− | * <math>\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}</math> | + | |

+ | Specifically, let <math>u, v : \mathbb{R \times R \to R}</math> be definted | ||

+ | by | ||

+ | <cmath> u(x,y) = \text{Re}\,f(x+iy), \qquad v(x,y) = \text{Im}\,f(x+iy) . </cmath> | ||

+ | If <math>z = x+iy</math>, then | ||

+ | <cmath> f(z) = u(x,y) + i v(x,y). </cmath> | ||

+ | |||

+ | It turns out that we can express the idea "<math>f</math> is holomorphic" | ||

+ | entirely in terms of partial derivatives of <math>u</math> and <math>v</math>. | ||

+ | |||

+ | '''Theorem.''' Let <math>D</math> be an [[open set | open]], [[connected]] [[subset]] of <math>\mathbb{C}</math>. | ||

+ | Let us abbreviate <math>x = \text{Re}\, z</math> and <math>y = \text{Im}\, z</math>. | ||

+ | Then the function <math>f</math> is holomorphic on <math>D</math> | ||

+ | if and only if all the [[partial derivative]]s of <math>u</math> and <math>v</math> with respect | ||

+ | to <math>x</math> and <math>y</math> are continuous on <math>D</math>, and the following system holds | ||

+ | for every point <math>z \in D</math>: | ||

+ | <cmath> \begin{align*} | ||

+ | \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} ,\\ | ||

+ | \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x}. | ||

+ | \end{align*} </cmath> | ||

+ | These equations are called the '''[[Cauchy-Riemann Equations]]'''. | ||

+ | |||

+ | For convenience, we may abbreviate | ||

+ | <cmath> \frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} | ||

+ | + i \frac{\partial v}{\partial x}, \qquad | ||

+ | \frac{\partial f}{\partial y} = \frac{\partial u}{\partial y} + | ||

+ | i \frac{\partial v}{\partial y} . </cmath> | ||

+ | With this abuse of notation, we may rewrite the Cauchy-Riemann | ||

+ | equations thus: | ||

+ | <cmath> \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} . </cmath> | ||

+ | |||

+ | === Proof of theorem === | ||

+ | First, suppose that <math>f</math> is | ||

+ | complex-differentiable at <math>z</math>. Then at <math>z</math>, | ||

+ | <cmath>\begin{align*} | ||

+ | \frac{\partial f}{\partial y} = \lim_{h\to 0} \frac{f(z+ih)-f(z)}{h} | ||

+ | &= i \cdot \lim_{h\to 0} \frac{f(z+ih) - f(z)}{ih} \\ | ||

+ | &= i \cdot f'(z) \\ | ||

+ | &= i \cdot \lim_{h\to 0} \frac{f(z+h)-f(z)}{h} | ||

+ | = i \cdot \frac{\partial f}{\partial x} . | ||

+ | \end{align*} | ||

+ | </cmath> | ||

+ | Breaking <math>f</math> into real and imaginary components, we see | ||

+ | <cmath> \frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} | ||

+ | = \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} | ||

+ | = -\frac{\partial v}{\partial x} + i \frac{\partial u}{\partial y}. </cmath> | ||

+ | Setting real and imaginary components equal, we obtain the | ||

+ | Cauchy-Riemann equations. It follows from the | ||

+ | [[Cauchy Integral Formula]] that the second derivative of <math>f</math> exists | ||

+ | at <math>z</math>; thus the derivative of <math>f</math> is continuous at <math>z</math>, and so | ||

+ | are the partial derivatives of <math>u</math> and <math>v</math>. | ||

− | + | Now, suppose the Cauchy-Riemann equations hold a point <math>z</math>, and | |

+ | that the partial derivatives of <math>u</math> and <math>v</math> exist and are continuous | ||

+ | in a [[neighborhood]] of <math>z</math>. Let <math>h = h_1 + i h_2</math> be an arbitrarily | ||

+ | small complex number, with <math>h_1, h_2 \in \mathbb{R}</math>. Then | ||

+ | <cmath> \begin{align*} | ||

+ | \frac{f(z + h) - f(z)}{h} &= \frac{f(z+h_1+ih_2)-f(z+h_1)}{h_1+ih_2} | ||

+ | + \frac{f(z+h_1)-f(z)}{h_1+ih_2} \\ | ||

+ | &\approx \frac{ih_2}{h_1+ih_2} \frac{\partial f}{\partial y}(z+h_1) | ||

+ | + \frac{h_1}{h_1 + ih_2} \frac{\partial f}{\partial x}(z) \\ | ||

+ | &\approx \frac{ih_2}{h_1+ih_2} \frac{\partial f}{\partial y}(z) | ||

+ | + \frac{h_1}{h_1 + ih_2} \frac{\partial f}{\partial x}(z) , | ||

+ | \end{align*} </cmath> | ||

+ | with the first approximation from the definition of the partial derivatives | ||

+ | and the second from the continuity of the partial derivatives. | ||

+ | We may force <math>h</math> to be small enough that both approximations | ||

+ | are arbitrarily accurate. Now, by the Cauchy-Riemann equations, | ||

+ | <cmath> \frac{i h_2}{h_1+ih_2} \frac{\partial f}{\partial y}(z) | ||

+ | + \frac{h_1}{h_1 + ih_2} \frac{\partial f}{\partial x}(z) | ||

+ | = \frac{\partial f}{\partial x} (z) .</cmath> | ||

+ | Therefore | ||

+ | <cmath> \lim_{h\to 0} \frac{f(z+h)-f(z)}{h} = \frac{\partial f}{\partial x} | ||

+ | (z) . </cmath> | ||

+ | In particular, the limit exists, so <math>f</math> is differentiable at <math>z</math>. | ||

+ | Since <math>z</math> was arbitrary, it follows that <math>f</math> is differentiable | ||

+ | everywhere in <math>D</math>. <math>\blacksquare</math> | ||

== Analytic Functions == | == Analytic Functions == | ||

− | A related notion to that of homolorphicity is that of analyticity. A function <math>f:\mathbb{C}\to\mathbb{C}</math> is said to be '''analytic''' at <math>z</math> if <math>f</math> has a convergent [[power series]] expansion on some [[neighborhood]] of <math>z</math>. Amazingly, it turns out that a function is holomorphic at <math>z</math> if and only if it is analytic at <math>z</math>. | + | A related notion to that of homolorphicity is that of analyticity. A |

+ | function <math>f:\mathbb{C}\to\mathbb{C}</math> is said to be '''analytic''' at | ||

+ | <math>z</math> if <math>f</math> has a convergent [[power series]] expansion on some | ||

+ | [[neighborhood]] of <math>z</math>. Amazingly, it turns out that a function is | ||

+ | holomorphic at <math>z</math> if and only if it is analytic at <math>z</math>. Furthermore, | ||

+ | its radius of convergence is the greatest lower bound of the distance | ||

+ | from <math>z</math> to a singularity. | ||

+ | |||

+ | This is not the case with real functions. Consider, for example, | ||

+ | the real function | ||

+ | <cmath> f(x) = \frac{1}{x^2 + 1} . </cmath> | ||

+ | It is infinitely differentiable along the entire real line, yet | ||

+ | its power series diverges when <math>\lvert x \rvert > 1</math>. But in the | ||

+ | complex plane we see that | ||

+ | <cmath> \frac{1}{z^2 +1} = \frac{1}{(z-i)(z+i)} </cmath> | ||

+ | has singularities at <math>z = \pm i</math>, so the power series must clearly | ||

+ | diverge when <math>\lvert z \rvert > 1</math>. | ||

+ | |||

+ | === Equivalence of Analytic and Holomorphic Functions === | ||

+ | |||

+ | We now prove that all holomorphic functions behave in this | ||

+ | orderly way. | ||

+ | |||

+ | '''Theorem.''' Let <math>D</math> be a connected, open subset of <math>\mathbb{C}</math>, | ||

+ | and let <math>f</math> be a holomorphic function on <math>D</math>. Then for any | ||

+ | <math>z_0 \in D</math>, the power series expansion of <math>f</math> aboud <math>z_0</math> converges, | ||

+ | and its radius of convergence is the greatest quantity <math>R</math> for which | ||

+ | there exists a holomorphic continuation of <math>f</math> to the set | ||

+ | <cmath> \{ z \in \mathbb{C} : \lvert z - z_0 \rvert < R \} . </cmath> | ||

+ | |||

+ | ''Proof.'' Since <math>z_0</math> is in <math>D</math>, there is some <math>R_0 >0</math> such | ||

+ | that <math>f</math> is holomorphic within <math>R_0+\epsilon</math> of <math>z</math>. Suppose that | ||

+ | <math>\lvert z - z_0 \rvert < R_0</math>. Let <math>C</math> be the simple, positively | ||

+ | oriented circle of radius <math>R_0</math> about <math>z_0</math>, and let <math>M</math> be an | ||

+ | upper bound on <math>\lvert f(z) \rvert</math> for <math>z\in C</math>. By the | ||

+ | [[Cauchy Integral Formula]], | ||

+ | <cmath> \left\lvert \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n \right\rvert | ||

+ | = \biggl\lvert \frac{(z-z_0)^n}{2\pi} \int\limits_C \frac{f(w)} | ||

+ | {(w-z_0)^{n+1}}dw \biggr\rvert | ||

+ | \le M \cdot \left\lvert \frac{z-z_0}{R_0} \right\rvert^n . </cmath> | ||

+ | The series thus converges geometrically. It follows that if | ||

+ | there is a holomorphic extension of <math>f</math> to the set | ||

+ | <cmath> \{ z \in \mathbb{C} : \lvert z \rvert < R \}, </cmath> | ||

+ | then the power series of <math>f</math> about <math>z</math> converges with radius at least | ||

+ | <math>R</math>. | ||

+ | |||

+ | Conversely, suppose that the power series expansion of <math>f</math> diverges | ||

+ | for at some <math>z</math> of distance less than <math>R</math> from <math>z_0</math>. Then | ||

+ | by the previous paragraph, there is no holomorphic extension of <math>f</math> | ||

+ | to all points of distance less than <math>R</math>. It follows that the | ||

+ | radius of convergence of the Taylor series expansion of <math>f</math> about | ||

+ | <math>z_0</math> is indeed the quantity as stated in the theorem. <math>\blacksquare</math> | ||

+ | |||

+ | === Strange Consequences of Extension === | ||

+ | |||

+ | In some cases, repeated extension of a function may lead | ||

+ | to bizarre consequences. For example, we may define a | ||

+ | square-root function <math>f(z)</math> that is holomorphic and defined | ||

+ | everywhere except on the set of non-positive real numbers. Any | ||

+ | power series expansion that avoids the origin will converge. However, | ||

+ | if we try to cross the negative real axis with a power series | ||

+ | expansion, we will find that our power series expansion gives | ||

+ | different results from our original function on the other side | ||

+ | of the axis! This is because the square root "function" is in | ||

+ | fact a [[multifunction]] that can restrict to a holomorphic | ||

+ | function on any open subset of <math>\mathbb{C}</math> that does not | ||

+ | include a closed path about the origin. | ||

+ | |||

+ | == See also == | ||

+ | |||

+ | * [[Cauchy Integral Formula]] | ||

+ | * [[Analytic function]] | ||

+ | * [[Taylor series]] | ||

+ | * [[Power series]] | ||

+ | * [[Derivative]] | ||

[[Category:Complex analysis]] | [[Category:Complex analysis]] |

## Latest revision as of 20:04, 26 July 2017

A **holomorphic function** is a
differentiable complex function. That is, just
as in the real case, is holomorphic at if
exists. This is much stronger
than in the real case since we must allow to approach zero from
any direction in the complex plane.

Usually, we speak of functions as holomorphic on (open) sets, rather than at points, for when we consider the behavior of a function at a point, we prefer to consider it in the context of the points nearby.

## Contents

## Cauchy-Riemann Equations

We can obtain an equivalent definition if we break and into real and imaginary components.

Specifically, let be definted by If , then

It turns out that we can express the idea " is holomorphic" entirely in terms of partial derivatives of and .

**Theorem.** Let be an open, connected subset of .
Let us abbreviate and .
Then the function is holomorphic on
if and only if all the partial derivatives of and with respect
to and are continuous on , and the following system holds
for every point :
These equations are called the **Cauchy-Riemann Equations**.

For convenience, we may abbreviate With this abuse of notation, we may rewrite the Cauchy-Riemann equations thus:

### Proof of theorem

First, suppose that is complex-differentiable at . Then at , Breaking into real and imaginary components, we see Setting real and imaginary components equal, we obtain the Cauchy-Riemann equations. It follows from the Cauchy Integral Formula that the second derivative of exists at ; thus the derivative of is continuous at , and so are the partial derivatives of and .

Now, suppose the Cauchy-Riemann equations hold a point , and that the partial derivatives of and exist and are continuous in a neighborhood of . Let be an arbitrarily small complex number, with . Then with the first approximation from the definition of the partial derivatives and the second from the continuity of the partial derivatives. We may force to be small enough that both approximations are arbitrarily accurate. Now, by the Cauchy-Riemann equations, Therefore In particular, the limit exists, so is differentiable at . Since was arbitrary, it follows that is differentiable everywhere in .

## Analytic Functions

A related notion to that of homolorphicity is that of analyticity. A
function is said to be **analytic** at
if has a convergent power series expansion on some
neighborhood of . Amazingly, it turns out that a function is
holomorphic at if and only if it is analytic at . Furthermore,
its radius of convergence is the greatest lower bound of the distance
from to a singularity.

This is not the case with real functions. Consider, for example, the real function It is infinitely differentiable along the entire real line, yet its power series diverges when . But in the complex plane we see that has singularities at , so the power series must clearly diverge when .

### Equivalence of Analytic and Holomorphic Functions

We now prove that all holomorphic functions behave in this orderly way.

**Theorem.** Let be a connected, open subset of ,
and let be a holomorphic function on . Then for any
, the power series expansion of aboud converges,
and its radius of convergence is the greatest quantity for which
there exists a holomorphic continuation of to the set

*Proof.* Since is in , there is some such
that is holomorphic within of . Suppose that
. Let be the simple, positively
oriented circle of radius about , and let be an
upper bound on for . By the
Cauchy Integral Formula,
The series thus converges geometrically. It follows that if
there is a holomorphic extension of to the set
then the power series of about converges with radius at least
.

Conversely, suppose that the power series expansion of diverges for at some of distance less than from . Then by the previous paragraph, there is no holomorphic extension of to all points of distance less than . It follows that the radius of convergence of the Taylor series expansion of about is indeed the quantity as stated in the theorem.

### Strange Consequences of Extension

In some cases, repeated extension of a function may lead to bizarre consequences. For example, we may define a square-root function that is holomorphic and defined everywhere except on the set of non-positive real numbers. Any power series expansion that avoids the origin will converge. However, if we try to cross the negative real axis with a power series expansion, we will find that our power series expansion gives different results from our original function on the other side of the axis! This is because the square root "function" is in fact a multifunction that can restrict to a holomorphic function on any open subset of that does not include a closed path about the origin.