5.3 Cauchy's integral formula

Cauchy's integral formula is a game-changer in complex analysis. It lets us find function values inside a contour using only the values on the boundary. This powerful tool simplifies complex integration and opens doors to evaluating tricky integrals.

Building on Cauchy's integral theorem, this formula connects analytic functions to their derivatives and integrals. It's the foundation for Taylor series expansions and residue calculations, making it essential for solving real-world problems in physics and engineering.

Cauchy's Integral Formula

Derivation from Cauchy's Integral Theorem

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• Cauchy's integral theorem states that if $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a simple closed contour in $D$, then $\oint_C f(z)dz = 0$
• To derive Cauchy's integral formula, consider a circle $C$ centered at $z_0$ with radius $r$, and a point $z$ inside $C$
  • Apply Cauchy's integral theorem to the function $f(\zeta)/(\zeta-z)$ on the domain $D - \{z\}$
  • Use the parametrization $\zeta = z_0 + re^{i\theta}$ and the fact that $\oint_C d\zeta/(\zeta-z) = 2\pi i$
  • Obtain the integral formula $f(z) = \frac{1}{2\pi i}\oint_C \frac{f(\zeta)}{\zeta-z}d\zeta$
• Cauchy's integral formula expresses the value of an analytic function at a point inside a contour in terms of the values of the function on the contour
  • Provides a powerful tool for evaluating complex integrals and derivatives

Relationship to Taylor Series

• Cauchy's integral formula can be used to derive the Taylor series expansion of an analytic function $f(z)$ around a point $z_0$
  • Differentiate the integral formula $n$ times to obtain $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(\zeta)}{(\zeta-z_0)^{n+1}}d\zeta$
  • Substitute the Taylor series of $f(\zeta)$ around $z_0$ into the integral and evaluate term by term
  • The resulting series is the Taylor series of $f(z)$ around $z_0$: $f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$
• The Taylor series provides a local approximation of an analytic function near a point
  • Useful for studying the behavior of functions and solving differential equations

Applications of Cauchy's Integral Formula

Evaluating Integrals and Derivatives

• To evaluate the integral of an analytic function $f(z)$ along a closed contour $C$, choose a point $z_0$ inside $C$ and apply Cauchy's integral formula: $\oint_C f(z)dz = 2\pi i \cdot f(z_0)$
  • Simplifies the calculation of complex integrals by reducing them to evaluating the function at a single point
• Cauchy's integral formula can be used to evaluate integrals of the form $\oint_C \frac{P(z)}{Q(z)}dz$, where $P$ and $Q$ are polynomials and $Q$ has simple zeros inside $C$
  • The result is a sum of residues at the zeros of $Q$, given by $\sum_{k=1}^n \frac{P(z_k)}{Q'(z_k)}$, where $z_k$ are the zeros of $Q$ inside $C$
• The derivative of an analytic function $f(z)$ can be computed using Cauchy's integral formula: $f'(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(\zeta)}{(\zeta-z_0)^2}d\zeta$
  • Higher-order derivatives can be obtained by differentiating under the integral sign: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(\zeta)}{(\zeta-z_0)^{n+1}}d\zeta$
  • Allows for the calculation of derivatives without the need for explicit differentiation

Residue Theorem

• The residue theorem is a powerful application of Cauchy's integral formula for evaluating integrals of the form $\oint_C \frac{f(z)}{g(z)}dz$, where $f$ and $g$ are analytic functions and $g$ has isolated zeros inside $C$
  • The residue of $f/g$ at a zero $z_0$ of $g$ is defined as $\text{Res}(f/g,z_0) = \frac{1}{(m-1)!}\lim_{z\to z_0} \frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^m\frac{f(z)}{g(z)}]$, where $m$ is the multiplicity of the zero
  • The residue theorem states that $\oint_C \frac{f(z)}{g(z)}dz = 2\pi i \sum_{k=1}^n \text{Res}(f/g,z_k)$, where $z_k$ are the zeros of $g$ inside $C$
• The residue theorem simplifies the evaluation of complex integrals by reducing them to the calculation of residues
  • Particularly useful for integrals involving rational functions, logarithms, and trigonometric functions
  • Finds applications in various fields, such as physics (Laplace transforms, Fourier analysis) and engineering (control theory, signal processing)

Properties of Analytic Functions

Mean Value Property

• The mean value property states that for an analytic function $f(z)$ in a disk $D(z_0,r)$, the value $f(z_0)$ is equal to the average of $f(z)$ over any circle $C$ centered at $z_0$ with radius $r$: $f(z_0) = \frac{1}{2\pi}\int_0^{2\pi} f(z_0+re^{i\theta})d\theta$
  • To prove the mean value property, apply Cauchy's integral formula to the circle $C$ and use the parametrization $z = z_0 + re^{i\theta}$
  • Demonstrates the smoothness and regularity of analytic functions
• The mean value property can be generalized to higher dimensions (harmonic functions) and non-circular domains (convex domains)
  • Plays a crucial role in the study of partial differential equations and potential theory

Maximum Modulus Principle

• Cauchy's integral formula can be used to prove the maximum modulus principle, which states that if $f(z)$ is analytic and non-constant in a domain $D$, then $|f(z)|$ cannot attain its maximum value at any interior point of $D$
  • If $|f(z)|$ attains its maximum at an interior point $z_0$, then by the mean value property, $f(z)$ must be constant in a neighborhood of $z_0$, contradicting the assumption that $f$ is non-constant
  • Implies that the maximum value of $|f(z)|$ must be attained on the boundary of $D$
• The maximum modulus principle has important consequences for the behavior of analytic functions
  • Liouville's theorem: A bounded entire function must be constant
  • Fundamental theorem of algebra: Every non-constant polynomial has a root
  • Open mapping theorem: A non-constant analytic function maps open sets to open sets

Uniqueness and Continuity

• Cauchy's integral formula implies the uniqueness of analytic functions: If two analytic functions $f(z)$ and $g(z)$ agree on a set with a limit point in a domain $D$, then they agree everywhere in $D$
  • Follows from the fact that the difference $f(z)-g(z)$ is analytic and vanishes on a set with a limit point, so it must be identically zero by the identity theorem
  • Allows for the extension of local properties of analytic functions to global properties
• Analytic functions are infinitely differentiable, and their derivatives are also analytic
  • The continuity and differentiability of analytic functions follow from the Cauchy-Riemann equations and the existence of the complex derivative
  • Higher-order derivatives can be computed using the generalized Cauchy integral formula: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(\zeta)}{(\zeta-z_0)^{n+1}}d\zeta$
  • Analytic functions possess a high degree of smoothness and regularity

Generalizations of Cauchy's Integral Formula

Higher-Order Derivatives and Integrals

• The generalized Cauchy integral formula expresses the $n$-th order derivative of an analytic function $f(z)$ at a point $z_0$ in terms of an integral: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(\zeta)}{(\zeta-z_0)^{n+1}}d\zeta$
  • Derived by differentiating Cauchy's integral formula $n$ times under the integral sign
  • Allows for the evaluation of higher-order derivatives without the need for explicit differentiation
• The generalized Cauchy integral formula can also be extended to integrate analytic functions: If $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a simple closed contour in $D$, then $\int_C f(z)dz = \frac{1}{2\pi i}\oint_C \frac{\int_\zeta f(\xi)d\xi}{\zeta-z}d\zeta$, where the inner integral is taken along a path from a fixed point to $\zeta$
  • Allows for the evaluation of integrals of analytic functions using contour integration techniques
  • Useful for computing definite integrals of real-valued functions by converting them to complex contour integrals (e.g., using the residue theorem)

Cauchy's Integral Formula for Unbounded Domains

• Cauchy's integral formula can be extended to unbounded domains by considering contours that extend to infinity
  • For an analytic function $f(z)$ in an unbounded domain $D$, choose a contour $C$ that consists of a large circle $C_R$ of radius $R$ and a path $\gamma$ connecting a point $z_0$ inside $C_R$ to infinity
  • Apply Cauchy's integral theorem to the region bounded by $C_R$ and $\gamma$, and take the limit as $R\to\infty$
  • The resulting formula is $f(z_0) = \frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)}{\zeta-z_0}d\zeta$, where the integral is taken along the path $\gamma$ from infinity to $z_0$
• The extension of Cauchy's integral formula to unbounded domains is particularly useful for studying the behavior of analytic functions at infinity
  • Allows for the classification of singularities (poles, essential singularities) and the computation of residues at infinity
  • Finds applications in complex analysis, such as the study of entire functions and meromorphic functions