Cauchy integral formula: Difference between revisions

Latest revision as of 18:43, 12 September 2008

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

For a circle with respect to any point in the interior

Suppose $U$ is a domain in $C$ and $f : U \to C$ is a holomorphic function. The Cauchy integral formula states that if $z_{0} \in U$ is such that the disk $D$ of radius $r$ about $z_{0}$ lies completely inside $U$ , and if $z$ is such that $| z - z_{0} | < r$ , we have:

$f (z) = \frac{1}{2 π i} \oint_{\partial D} \frac{f (ξ)}{ξ - z} d ξ$

In other words, we can determine the value of $f$ at any point in the interior, by knowing its value at all points on the boundary.

For any union of piecewise smooth curves and any point

Further information: Winding number version of Cauchy integral formula

Related facts

Facts used

Cauchy integral formula for constant functions: In other words, the fact that for a constant function, the Cauchy integral formula holds.
A version of Fubini's theorem, that allows the exchange of a real and a complex integral when the total quantities are absolutely integrable.
fundamental theorem of complex calculus.

Proof

Given: $U$ is a domain in $C$ and $f : U \to C$ is a holomorphic function. $z_{0} \in U$ is such that the disk $D$ of radius $r$ about $z_{0}$ lies completely inside $U$ , and $z$ is such that $| z - z_{0} | < r$ .

To prove:

$f (z) = \frac{1}{2 π i} \oint_{\partial D} \frac{f (ξ)}{ξ - z} d ξ$

Proof: The proof proceeds in two steps. We first use the Cauchy integral formula for constant functions (fact (1)) to show that for the constant function $w \mapsto f (z)$ , we have:

$f (z) = \oint_{\partial D} \frac{f (z)}{ξ - z} d ξ$

We then show that:

$\oint_{\partial D} \frac{f (ξ)}{ξ - z} d ξ = \oint_{\partial D} \frac{f (z)}{ξ - z} d ξ$ .

Let's do the second step now. We do this by a homotopy method. Namely, consider the function $[0, 1] \times \bar{D} ∖ {z} \to C$ :

$F (t, w) = \frac{f ((1 - t) z + t w)}{w - z}$ .

Clearly, $F$ is holomorphic in $w$ , and we have:

$\frac{\partial F}{\partial t} = f^{'} ((1 - t) z + t w)$ .

This yields, by fact (3), that for any $t \in [0, 1]$ :

$\oint_{\partial D} \frac{\partial F (t, ξ)}{\partial t} d ξ = 0$

Integrating over $t \in [0, 1]$ yields:

$\int_{0}^{1} \oint_{\partial D} \frac{\partial F (t, ξ)}{\partial t} d ξ d t = 0$

We now exchange the two integrals by fact (2). We get:

$\oint_{\partial D} \int_{0}^{1} \frac{\partial F (t, ξ)}{\partial t} d t d ξ = 0$ .

Next, we apply the fundamental theorem of real calculus to evaluate the inner integral, and obtain:

$\oint_{\partial D} F (1, ξ) - F (0, ξ) d ξ = 0$ .

Plugging in values yields:

$\oint_{\partial D} \frac{f (z) - f (ξ)}{z - ξ} = 0$

which, on rearrangement, gives the desired equality:

$\oint_{\partial D} \frac{f (ξ)}{ξ - z} d ξ = \oint_{\partial D} \frac{f (z)}{ξ - z} d ξ$ .

@@ Line 10: / Line 10: @@
 In other words, we can determine the value of <math>f</math> at ''any point in the interior'', by knowing its value at all points on the boundary.
-===For any simple closed curve and any point in the interior===
+===For any union of piecewise smooth curves and any point===
+{{further|[[Winding number version of Cauchy integral formula]]}}
+==Related facts==
+* [[Cauchy integral formula for derivatives]]
+* [[Winding number version of Cauchy integral formula]]
+* [[Homotopy-invariance formulation of Cauchy's theorem]]
+==Facts used==
+# [[Cauchy integral formula for constant functions]]: In other words, the fact that for a constant function, the Cauchy integral formula holds.
+# A version of Fubini's theorem, that allows the exchange of a real and a complex integral when the total quantities are absolutely integrable.
+# [[uses::fundamental theorem of complex calculus]].
+==Proof==
+'''Given''': <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. <math>z_0 \in U</math> is such that the disk <math>D</math> of radius <math>r</math> about <math>z_0</math> lies completely inside <math>U</math>, and <math>z</math> is such that <math>|z - z_0| < r</math>.
+'''To prove''':
+<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi</math>
+'''Proof''': The proof proceeds in two steps. We first use the Cauchy integral formula for ''constant'' functions (fact (1)) to show that for the constant function <math>w \mapsto f(z)</math>, we have:
+<math>f(z) = \oint_{\partial D} \frac{f(z)}{\xi - z} \, d\xi</math>
+We then show that:
+<math> \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi = \oint_{\partial D} \frac{f(z)}{\xi - z} \, d\xi</math>.
+Let's do the second step now. We do this by a ''homotopy'' method. Namely, consider the function <math>[0,1] \times \overline{D} \setminus \{ z \} \to \mathbb{C}</math>:
+<math>F(t,w) = \frac{f((1-t)z + tw)}{w - z}</math>.
+Clearly, <math>F</math> is holomorphic in <math>w</math>, and we have:
+<math>\frac{\partial F}{\partial t} = f'((1 - t)z + tw)</math>.
-This is the same integral formula, this time applied to ''any'' simple closed curve and any point in the interior.
+This yields, by fact (3), that for any <math>t \in [0,1]</math>:
-Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. The '''Cauchy integral formula''' states that if <math>\gamma</math> is a smooth simple closed curve lying completely inside <math>U</math>, and <math>z_0</math> is in the component of the complement of <math>\gamma</math> that lies completely inside <math>U</math>, then we have:
+<math>\oint_{\partial D} \frac{\partial F(t,\xi)}{\partial t} \, d\xi = 0</math>
-<math>f(z_0) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(z)}{z - z_0} \, dz</math>
+Integrating over <math>t \in [0,1]</math> yields:
-In other words, the curve doesn't need to be a circle and the point can be anywhere inside.
+<math>\int_0^1 \oint_{\partial D} \frac{\partial F(t,\xi)}{\partial t} \, d\xi \, dt = 0</math>
-===For any union of piecewise smooth curves and any point===
+We now exchange the two integrals by fact (2). We get:
-{{further|[[Winding number version of Cauchy integral formula]]}}
+<math>\oint_{\partial D} \int_0^1 \frac{\partial F(t,\xi)}{\partial t} \, dt \, d\xi = 0</math>.
-Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]].
+Next, we apply the fundamental theorem of ''real'' calculus to evaluate the inner integral, and obtain:
-Suppose <math>c</math> is a sum of oriented piecewise smooth loops <math>c_1, c_2, \ldots, c_r</math>, each completely inside <math>U</math>. In other words, when we use the symbol <math>\int_c g(z) dz</math> we actually mean <math>\sum_1^r \int_{c_r} g(z) dz</math>.
+<math>\oint_{\partial D} F(1,\xi) - F(0,\xi) \, d\xi = 0</math>.
-Denote by <math>n(c;z_0)</math> the sum of the winding numbers of the <math>c_k</math> about <math>z_0</math>. Suppose <math>c</math> is zero-homologous, i.e. <math>n(c;z) = 0</math> for <math>z \in \mathbb{C}\setminus U</math>. Then we have, for any <math>z_0 \in U</math>:
+Plugging in values yields:
-<math>n(c;z_0)f(z_0) = \frac{1}{2 \pi i} \int_c \frac{f(z)}{z - z_0} \, dz</math>
+<math>\oint_{\partial D} \frac{f(z) - f(\xi)}{z - \xi} = 0</math>
-==Related facts==
+which, on rearrangement, gives the desired equality:
-* [[Cauchy integral formula for derivatives]]
+<math> \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi = \oint_{\partial D} \frac{f(z)}{\xi - z} \, d\xi</math>.
-* [[Winding number version of Cauchy integral formula]]
-* [[Homotopy-invariance formulation of Cauchy's theorem]]