4.3 Cauchy's integral formula

Cauchy's integral formula is a game-changer in complex analysis. It links an analytic function's values inside a closed contour to its values on the contour itself. This powerful tool opens doors to computing complex integrals and understanding analytic functions' properties.

The formula assumes an analytic function in a simply connected domain and a simple closed contour within it. It's expressed as $f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-z_0}dz$, where $z_0$ is inside the contour $C$.

Definition of Cauchy's integral formula

• Cauchy's integral formula is a fundamental result in complex analysis that relates the values of an analytic function inside a simple closed contour to the values of the function on the contour itself
• Establishes a connection between the behavior of a complex function inside a region and its values on the boundary of that region
• Allows for the computation of complex integrals and the derivation of various properties of analytic functions

Assumptions for Cauchy's integral formula

• The function $f(z)$ must be analytic in a simply connected domain $D$
• The contour $C$ must be a simple closed contour that lies entirely within the domain $D$
• The point $z_0$ at which the value of $f(z)$ is to be evaluated must lie inside the contour $C$

Components of Cauchy's integral formula

Analytic functions in Cauchy's integral formula

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• Analytic functions are complex functions that are differentiable at every point in their domain
• Cauchy's integral formula applies specifically to analytic functions, as they possess certain properties that allow for the formula to hold
• Examples of analytic functions include polynomials, exponential functions, and trigonometric functions

Simple closed contours in Cauchy's integral formula

• A simple closed contour is a closed curve in the complex plane that does not intersect itself and has a well-defined interior and exterior
• Cauchy's integral formula requires the contour of integration to be a simple closed contour
• Examples of simple closed contours include circles, rectangles, and triangles

Contour integrals in Cauchy's integral formula

• Contour integrals are integrals of complex functions along a given contour in the complex plane
• Cauchy's integral formula expresses the value of an analytic function at a point inside a contour in terms of a contour integral along the boundary
• The contour integral in Cauchy's formula is given by $\frac{1}{2\pi i}\oint_C \frac{f(z)}{z-z_0}dz$, where $C$ is the contour and $z_0$ is the point inside the contour

Derivation of Cauchy's integral formula

Cauchy-Goursat theorem

• The Cauchy-Goursat theorem states that if a function is analytic in a simply connected domain, then the contour integral of the function along any closed contour within the domain is zero
• This theorem is a crucial step in the derivation of Cauchy's integral formula, as it allows for the deformation of contours without changing the value of the integral

Deformation of contours

• The deformation of contours is a technique used in the derivation of Cauchy's integral formula
• By deforming the contour of integration to a circle centered at the point $z_0$, the integral can be simplified using the Cauchy-Goursat theorem
• The deformation of contours is possible due to the analyticity of the function and the simply connected nature of the domain

Proof of Cauchy's integral formula

• The proof of Cauchy's integral formula involves the use of the Cauchy-Goursat theorem and the deformation of contours
• By considering a small circle centered at the point $z_0$ and applying the Cauchy-Goursat theorem, the integral can be evaluated using the residue theorem
• The result is then obtained by taking the limit as the radius of the circle approaches zero, yielding the Cauchy's integral formula: $f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-z_0}dz$

Consequences of Cauchy's integral formula

Analyticity and differentiability

• Cauchy's integral formula implies that if a function is analytic in a domain, then it is infinitely differentiable in that domain
• The derivatives of an analytic function can be obtained by differentiating under the integral sign in Cauchy's formula
• This result establishes a strong connection between analyticity and differentiability in complex analysis

Higher-order derivatives

• Cauchy's integral formula can be used to derive expressions for higher-order derivatives of analytic functions
• By differentiating the Cauchy's integral formula multiple times with respect to $z_0$, one obtains formulas for the derivatives of $f(z)$ at $z_0$
• The $n$-th order derivative of $f(z)$ at $z_0$ is given by $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z-z_0)^{n+1}}dz$

Cauchy's integral formulas for derivatives

• Cauchy's integral formulas for derivatives are a set of formulas that express the derivatives of an analytic function in terms of contour integrals
• These formulas are obtained by differentiating Cauchy's integral formula multiple times with respect to the point $z_0$
• Cauchy's integral formulas for derivatives provide a powerful tool for computing derivatives of complex functions and studying their properties

Applications of Cauchy's integral formula

Evaluation of integrals

• Cauchy's integral formula can be used to evaluate certain types of complex integrals
• By choosing an appropriate contour and applying Cauchy's formula, one can often simplify the evaluation of integrals that would be difficult to compute directly
• Examples include integrals of rational functions and integrals involving exponential or trigonometric functions

Calculation of Taylor series

• Cauchy's integral formula can be used to derive the Taylor series expansion of an analytic function
• By expanding the integrand in Cauchy's formula in a geometric series and interchanging the sum and the integral, one obtains the Taylor series coefficients in terms of contour integrals
• This method provides an alternative approach to computing Taylor series expansions and studying their convergence properties

Residue theorem and its applications

• The residue theorem is a powerful result in complex analysis that relates the contour integral of a meromorphic function to the sum of its residues
• Cauchy's integral formula plays a crucial role in the proof of the residue theorem, as it allows for the evaluation of integrals around singularities
• The residue theorem has numerous applications, such as evaluating real integrals, summing series, and studying the behavior of complex functions near singularities

Generalizations of Cauchy's integral formula

Cauchy's integral formula for multiply connected domains

• Cauchy's integral formula can be generalized to multiply connected domains, which are domains with holes or cut
• In this case, the formula involves a sum of contour integrals along the boundaries of the domain, with appropriate coefficients determined by the topology of the domain
• This generalization allows for the application of Cauchy's integral formula to a wider class of domains and functions

Cauchy's integral formula for meromorphic functions

• Cauchy's integral formula can also be extended to meromorphic functions, which are functions that are analytic except at a finite number of isolated singularities (poles)
• The formula for meromorphic functions includes additional terms that account for the contributions of the poles
• This generalization is particularly useful in the study of rational functions and their residues

Relationship to other theorems

Cauchy's integral formula vs Morera's theorem

• Morera's theorem is a converse of Cauchy's integral theorem, stating that if a continuous function satisfies $\oint_C f(z)dz = 0$ for every closed contour $C$ in a domain, then $f(z)$ is analytic in that domain
• While Cauchy's integral formula allows for the computation of the values of an analytic function from its boundary values, Morera's theorem provides a criterion for a function to be analytic based on its contour integrals

Cauchy's integral formula vs Liouville's theorem

• Liouville's theorem states that if a function is analytic and bounded in the entire complex plane, then it must be a constant
• Cauchy's integral formula can be used to prove Liouville's theorem by showing that the derivatives of a bounded analytic function are all zero
• Both theorems highlight the strong rigidity of analytic functions and their behavior in the complex plane

Examples and exercises

Evaluating integrals using Cauchy's integral formula

• Example: Evaluate the integral $\oint_C \frac{\sin(z)}{z-\pi}dz$, where $C$ is the circle $|z|=2$
• Solution: By Cauchy's integral formula, the integral equals $2\pi i \sin(\pi) = 0$, since $\pi$ lies inside the contour and $\sin(\pi)=0$

Proving properties of analytic functions

• Exercise: Use Cauchy's integral formula to prove that if $f(z)$ is analytic in a domain $D$ and $f(z)=0$ for all $z$ on a simple closed contour $C$ within $D$, then $f(z)\equiv 0$ in the interior of $C$
• Hint: Apply Cauchy's integral formula to express $f(z)$ at any point inside $C$ in terms of its values on the contour, and use the given condition to conclude the result