Cauchy's Integral Formula is a cornerstone of complex analysis, stating that a holomorphic (analytic) function inside a closed loop (contour) is entirely determined by its values on that boundary, allowing calculation of function values and derivatives at any interior point using a contour integral, often expressed as \(f(z_{0})=\frac{1}{2\pi i}\oint _{C}\frac{f(z)}{z-z_{0}}dz\) for the function value, and \(f^{(n)}(z_{0})=\frac{n!}{2\pi i}\oint _{C}\frac{f(z)}{(z-z_{0})^{n+1}}dz\) for its \(n\)-th derivative, revealing differentiation is like integration for analytic functions.
\[ f(z_0) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - z_0} \, dz \]
\[ g(z) = \frac{f(z)}{z - z_0} \]
The function \( f(z) \) is analytic everywhere inside \( C \), so \( g(z) \) is analytic everywhere inside \( C \) except at the singular point \( z = z_0 \).
We enclose the point \( z_0 \) with a small circle \( C_{\epsilon} \) centered at \( z_0 \) with radius \( \epsilon \), lying entirely inside \( C \). By the Principle of Deformation of Contours (a consequence of the Cauchy-Goursat Theorem), the integral over the outer boundary \( C \) is equal to the integral over the small inner circle \( C_{\epsilon} \):
\[ \oint_{C} \frac{f(z)}{z - z_0} \, dz = \oint_{C_{\epsilon}} \frac{f(z)}{z - z_0} \, dz \]
\[ \oint_{C_{\epsilon}} \frac{f(z)}{z - z_0} \, dz = \oint_{C_{\epsilon}} \frac{f(z_0) + (f(z) - f(z_0))}{z - z_0} \, dz \]
This splits into two separate integrals:
\[ I = f(z_0) \oint_{C_{\epsilon}} \frac{1}{z - z_0} \, dz + \oint_{C_{\epsilon}} \frac{f(z) - f(z_0)}{z - z_0} \, dz \]
\[ \oint_{C_{\epsilon}} \frac{1}{z - z_0} \, dz = \int_{0}^{2\pi} \frac{1}{\epsilon e^{i\theta}} (i \epsilon e^{i\theta}) \, d\theta = \int_{0}^{2\pi} i \, d\theta = 2\pi i \]
Therefore, the first term is:
\[ f(z_0) \cdot (2\pi i) \]
Using the ML Inequality (Estimation Lemma):
\[ |J| \le \max_{z \in C_{\epsilon}} \left| \frac{f(z) - f(z_0)}{z - z_0} \right| \cdot \text{Length}(C_{\epsilon}) \]
\[ |J| \le \frac{\delta}{\epsilon} \cdot 2\pi\epsilon = 2\pi\delta \]
As \( \epsilon \to 0 \), we have \( \delta \to 0 \), so the integral \( J \) vanishes.
\[ \oint_{C} \frac{f(z)}{z - z_0} \, dz = 2\pi i f(z_0) \]
Rearranging yields the formula:
\[ f(z_0) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - z_0} \, dz \]
\[ \blacksquare \]
Theorem Statement:
Let \( f(z) \) be analytic in a simply connected domain \( D \). Let \( C \) be a simple closed counter-clockwise contour lying entirely within \( D \). If \( z_0 \) is any point interior to \( C \), then:\[ f(z_0) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - z_0} \, dz \]
Proof
1. Setup and Deformation
Consider the function defined by the integrand:\[ g(z) = \frac{f(z)}{z - z_0} \]
The function \( f(z) \) is analytic everywhere inside \( C \), so \( g(z) \) is analytic everywhere inside \( C \) except at the singular point \( z = z_0 \).
We enclose the point \( z_0 \) with a small circle \( C_{\epsilon} \) centered at \( z_0 \) with radius \( \epsilon \), lying entirely inside \( C \). By the Principle of Deformation of Contours (a consequence of the Cauchy-Goursat Theorem), the integral over the outer boundary \( C \) is equal to the integral over the small inner circle \( C_{\epsilon} \):
\[ \oint_{C} \frac{f(z)}{z - z_0} \, dz = \oint_{C_{\epsilon}} \frac{f(z)}{z - z_0} \, dz \]
2. Manipulating the Integral
We can rewrite the integrand on the right-hand side by adding and subtracting \( f(z_0) \) in the numerator:\[ \oint_{C_{\epsilon}} \frac{f(z)}{z - z_0} \, dz = \oint_{C_{\epsilon}} \frac{f(z_0) + (f(z) - f(z_0))}{z - z_0} \, dz \]
This splits into two separate integrals:
\[ I = f(z_0) \oint_{C_{\epsilon}} \frac{1}{z - z_0} \, dz + \oint_{C_{\epsilon}} \frac{f(z) - f(z_0)}{z - z_0} \, dz \]
3. Evaluating the First Term
On the circle \( C_{\epsilon} \), we parameterize \( z \) as \( z = z_0 + \epsilon e^{i\theta} \), where \( 0 \le \theta \le 2\pi \). Thus, \( dz = i \epsilon e^{i\theta} d\theta \).\[ \oint_{C_{\epsilon}} \frac{1}{z - z_0} \, dz = \int_{0}^{2\pi} \frac{1}{\epsilon e^{i\theta}} (i \epsilon e^{i\theta}) \, d\theta = \int_{0}^{2\pi} i \, d\theta = 2\pi i \]
Therefore, the first term is:
\[ f(z_0) \cdot (2\pi i) \]
4. Bounding the Second Term (The Error)
Let \( J = \oint_{C_{\epsilon}} \frac{f(z) - f(z_0)}{z - z_0} \, dz \). Since \( f(z) \) is continuous at \( z_0 \), for any \( \delta > 0 \), we can choose \( \epsilon \) small enough such that \( |f(z) - f(z_0)| < \delta \).Using the ML Inequality (Estimation Lemma):
\[ |J| \le \max_{z \in C_{\epsilon}} \left| \frac{f(z) - f(z_0)}{z - z_0} \right| \cdot \text{Length}(C_{\epsilon}) \]
\[ |J| \le \frac{\delta}{\epsilon} \cdot 2\pi\epsilon = 2\pi\delta \]
As \( \epsilon \to 0 \), we have \( \delta \to 0 \), so the integral \( J \) vanishes.
Conclusion
Since the value of the original integral is independent of \( \epsilon \), taking the limit as \( \epsilon \to 0 \) gives:\[ \oint_{C} \frac{f(z)}{z - z_0} \, dz = 2\pi i f(z_0) \]
Rearranging yields the formula:
\[ f(z_0) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z - z_0} \, dz \]
\[ \blacksquare \]