Difference between revisions of "Holomorphic function"
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entirely in terms of partial derivatives of <math>u</math> and <math>v</math>. | entirely in terms of partial derivatives of <math>u</math> and <math>v</math>. | ||
− | '''Theorem.''' Let <math>D</math> be an open, connected subset of <math>\mathbb{C}</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>. | 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> | Then the function <math>f</math> is holomorphic on <math>D</math> | ||
− | if and only if all the partial | + | 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 | 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>: | for every point <math>z \in D</math>: | ||
Line 46: | Line 46: | ||
<cmath> \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} . </cmath> | <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>, | complex-differentiable at <math>z</math>. Then at <math>z</math>, | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
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&= i \cdot f'(z) \\ | &= i \cdot f'(z) \\ | ||
&= i \cdot \lim_{h\to 0} \frac{f(z+h)-f(z)}{h} | &= i \cdot \lim_{h\to 0} \frac{f(z+h)-f(z)}{h} | ||
− | = i \cdot \frac{\partial f}{\partial x} . </cmath> | + | = i \cdot \frac{\partial f}{\partial x} . |
+ | \end{align*} | ||
+ | </cmath> | ||
Breaking <math>f</math> into real and imaginary components, we see | Breaking <math>f</math> into real and imaginary components, we see | ||
<cmath> \frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} | <cmath> \frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} | ||
Line 96: | Line 99: | ||
<math>z</math> if <math>f</math> has a convergent [[power series]] expansion on some | <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 | [[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>. | + | 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 21:04, 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.