Difference between revisions of "Cauchy's Integral Formula"
(added relation for nth derivatives) |
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Let <math>D</math> denote the interior of the region bounded by <math>C</math>. | Let <math>D</math> denote the interior of the region bounded by <math>C</math>. | ||
− | Let <math>C_r</math> denote a simple counterclockwise | + | Let <math>C_r</math> denote a simple counterclockwise circle about <math>z_0</math> |
of radius <math>r</math>. Since the interior of the region bounded by <math>C</math> | of radius <math>r</math>. Since the interior of the region bounded by <math>C</math> | ||
is an [[open set]], there is some <math>R</math> such that <math>C_r \subset D</math> | is an [[open set]], there is some <math>R</math> such that <math>C_r \subset D</math> | ||
Line 48: | Line 48: | ||
By induction, we see that the <math>n</math>th derivative of <math>f</math> at <math>z_0</math> is | By induction, we see that the <math>n</math>th derivative of <math>f</math> at <math>z_0</math> is | ||
− | <cmath> f^{(n)}(z_0) = | + | <cmath> f^{(n)}(z_0) = \frac{n!}{2\pi i} \int\limits_C |
+ | \frac{f(z)}{(z-z_0)^{n+1}}dz, </cmath> | ||
for <math>n>0</math>. In particular, the <math>n</math>th derivative ''exists'' at <math>z_0</math>, | for <math>n>0</math>. In particular, the <math>n</math>th derivative ''exists'' at <math>z_0</math>, | ||
for all <math>n>0</math>. In other words, if a function <math>f</math> is | for all <math>n>0</math>. In other words, if a function <math>f</math> is | ||
complex-differentiable on some region, then it is ''infinitely | complex-differentiable on some region, then it is ''infinitely | ||
differentiable'' on the interior of that region. | differentiable'' on the interior of that region. | ||
+ | |||
+ | Since the <math>(n+1)</math>th derivative exists in general, it follows that | ||
+ | the <math>n</math>th derivative is continuous. This is not true for functions | ||
+ | of real variables! For instance the real function | ||
+ | <cmath> f(x) = \begin{cases} x \sin(1/x), & x \neq 0 \\ 0, & x=0 \end{cases} </cmath> | ||
+ | is everywhere differentiable, but its derivative is mysteriously | ||
+ | not continuous at <math>x=0</math>. In complex analysis, the mystery disappears: | ||
+ | the function <math>z\sin(1/z) = z\frac{e^{i/z} - e^{-i/z}}{2i}</math> has an | ||
+ | [[essential singularity]] at <math>z=0</math>, so we can't establish a derivative | ||
+ | there in any case. | ||
+ | |||
+ | The theorem is useful for estimating a function (or its <math>n</math>th derivative) | ||
+ | at a point based on the behavior of the function around the point. | ||
+ | For instance, the theorem yields an easy proof that [[holomorphic function]]s | ||
+ | are in fact [[analytic function | analytic]]. | ||
== See also == | == See also == |
Latest revision as of 17:19, 18 January 2024
Cauchy's Integral Formula is a fundamental result in
complex analysis. It states that if is a subset of
the complex plane containing a simple counterclockwise loop
and
the region bounded by
, and
is a complex-differentiable function on
, then for any
in the interior of the region bounded by
,
Proof
Let denote the interior of the region bounded by
.
Let
denote a simple counterclockwise circle about
of radius
. Since the interior of the region bounded by
is an open set, there is some
such that
for all
. For such values of
,
by application of Cauchy's Integral Theorem.
Since is differentiable at
, for any
we
may pick an arbitarily small
such that
whenever
. Let us parameterize
as
, for
. Since
(again by Cauchy's Integral Theorem), it follows
that
Since
and
can simultaneously become arbitrarily
small, it follows that
which is equivalent to the desired theorem.
Consequences
By induction, we see that the th derivative of
at
is
for
. In particular, the
th derivative exists at
,
for all
. In other words, if a function
is
complex-differentiable on some region, then it is infinitely
differentiable on the interior of that region.
Since the th derivative exists in general, it follows that
the
th derivative is continuous. This is not true for functions
of real variables! For instance the real function
is everywhere differentiable, but its derivative is mysteriously
not continuous at
. In complex analysis, the mystery disappears:
the function
has an
essential singularity at
, so we can't establish a derivative
there in any case.
The theorem is useful for estimating a function (or its th derivative)
at a point based on the behavior of the function around the point.
For instance, the theorem yields an easy proof that holomorphic functions
are in fact analytic.