Difference between revisions of "Cauchy's Integral Formula"

Latest revision as of 16:19, 18 January 2024

Cauchy's Integral Formula is a fundamental result in complex analysis. It states that if $U$ is a subset of the complex plane containing a simple counterclockwise loop $C$ and the region bounded by $C$ , and $f$ is a complex-differentiable function on $U$ , then for any $z_0$ in the interior of the region bounded by $C$ , $\frac{1}{2\pi i} \int\limits_C \frac{f(z)}{z- z_0}dz = f(z_0) .$

Proof

Let $D$ denote the interior of the region bounded by $C$ . Let $C_r$ denote a simple counterclockwise circle about $z_0$ of radius $r$ . Since the interior of the region bounded by $C$ is an open set, there is some $R$ such that $C_r \subset D$ for all $r \in (0, R)$ . For such values of $r$ , $\int\limits_C \frac{f(z)}{z-z_0}dz = \int\limits_{C_r} \frac{f(z)}{z-z_0}dz ,$ by application of Cauchy's Integral Theorem.

Since $f$ is differentiable at $z_0$ , for any $\epsilon$ we may pick an arbitarily small $r>0$ such that $\left\lvert \frac{f(z)-f(z_0)}{z-z_0} - f'(z_0) \right\rvert < \epsilon$ whenever $\lvert z - z_0 \rvert \le r$ . Let us parameterize $C_r$ as $h(t) = r e^{it}+ z_0$ , for $t\in [0,2\pi]$ . Since $\int\limits_{C_r} f'(z_0)dz = 0$ (again by Cauchy's Integral Theorem), it follows that $\begin{align*} \biggl\lvert \int\limits_{C_r} \frac{f(z)}{z-z_0}dz - \int\limits_{C_r} \frac{f(z_0)}{z-z_0}dz \biggr\rvert &= \biggl\lvert \int\limits_{C_r} \left[ \frac{f(z) - f(z_0)}{z-z_0} - f'(z_0) \right] dz \biggr\rvert \\ &\le \int\limits_0^{2\pi} \left\lvert \frac{f(h(t)) - f(z_0)}{ h(t) - z_0} - f'(z_0) \right\vert \cdot r dt \\ &< \int\limits_0^{2\pi} \epsilon \cdot r dt = 2\pi \epsilon r . \end{align*}$ Since $\epsilon$ and $r$ can simultaneously become arbitrarily small, it follows that $\begin{align*} \int\limits_C \frac{f(z)}{z-z_0}dz &= \int\limits_{C_r} \frac{f(z_0)} {z- z_0}dz \\ &= f(z_0) \int\limits_{0}^{2\pi} \frac{h'(t)}{h(t) - z_0} dt \\ &= f(z_0) \int\limits_{0}^{2\pi} \frac{ir e^{it}}{re^{it}} dt \\ &= f(z_0) \cdot 2\pi i , \end{align*}$ which is equivalent to the desired theorem. $\blacksquare$

Consequences

By induction, we see that the $n$ th derivative of $f$ at $z_0$ is $f^{(n)}(z_0) = \frac{n!}{2\pi i} \int\limits_C \frac{f(z)}{(z-z_0)^{n+1}}dz,$ for $n>0$ . In particular, the $n$ th derivative exists at $z_0$ , for all $n>0$ . In other words, if a function $f$ is complex-differentiable on some region, then it is infinitely differentiable on the interior of that region.

Since the $(n+1)$ th derivative exists in general, it follows that the $n$ th derivative is continuous. This is not true for functions of real variables! For instance the real function $f(x) = \begin{cases} x \sin(1/x), & x \neq 0 \\ 0, & x=0 \end{cases}$ is everywhere differentiable, but its derivative is mysteriously not continuous at $x=0$ . In complex analysis, the mystery disappears: the function $z\sin(1/z) = z\frac{e^{i/z} - e^{-i/z}}{2i}$ has an essential singularity at $z=0$ , so we can't establish a derivative there in any case.

The theorem is useful for estimating a function (or its $n$ 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.

@@ Line 9: / Line 9: @@
 Let <math>D</math> denote the interior of the region bounded by <math>C</math>.
-Let <math>C_r</math> denote a simple counterclockwise loop about <math>z_0</math>
+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>
 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
-<cmath> f^{(n)}(z_0) = (-1)^{n-1} (n-1)! \int\limits_C \frac{f(z)}{(z-z_0)^n}dz, </cmath>
+<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 all <math>n>0</math>.  In other words, if a function <math>f</math> is
 complex-differentiable on some region, then it is ''infinitely
 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 ==

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Difference between revisions of "Cauchy's Integral Formula"

Latest revision as of 16:19, 18 January 2024

Proof

Consequences

See also