8.3 Cauchy's integral formula

Cauchy's integral formula is a powerful tool in complex analysis. It lets us calculate function values inside closed contours using integrals. This formula connects differentiation and integration in the complex plane, making it crucial for solving tricky problems.

The formula has many applications, from evaluating real integrals to proving the fundamental theorem of algebra. It's essential for understanding how complex functions behave and for tackling advanced topics in mathematics and physics.

Cauchy's Integral Formulas

Cauchy's Integral Formula and Its Applications

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• Cauchy's integral formula expresses the value of an analytic function $[f(z)](https://www.fiveableKeyTerm:f(z))$ inside a closed contour $C$ as a contour integral: $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz$
• Applies to functions that are analytic (differentiable) everywhere inside and on the contour $C$
• Enables the computation of complex integrals by reducing them to a sum of residues (residue theorem)
• Useful in evaluating definite integrals of real-valued functions by converting them into complex contour integrals (e.g., trigonometric integrals, improper integrals)

Derivatives and the Fundamental Theorem of Algebra

• Cauchy's integral formula for derivatives expresses the $n$-th derivative of an analytic function $f(z)$ as: $f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} dz$
• Proves that analytic functions have derivatives of all orders
• The fundamental theorem of algebra states that every non-constant polynomial $p(z)$ of degree $n$ has exactly $n$ complex roots (counting multiplicities)
• Can be proven using Cauchy's integral formula and the maximum modulus principle (e.g., by showing that $1/p(z)$ must have a pole inside any sufficiently large contour)

Complex Analysis Theorems

Liouville's Theorem and the Maximum Modulus Principle

• Liouville's theorem states that every bounded entire function (analytic on the whole complex plane) must be constant
• Proves that certain functions (e.g., $e^z$, $\sin(z)$) cannot be bounded on the entire complex plane
• The maximum modulus principle asserts that if $f(z)$ is analytic in a region $R$ and continuous on its boundary, then $|f(z)|$ attains its maximum value on the boundary of $R$
• Implies that a non-constant analytic function cannot have a local maximum or minimum inside its domain

The Residue Theorem and Its Applications

• The residue theorem relates the contour integral of a meromorphic function $f(z)$ (analytic except for poles) to the sum of its residues: $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$
• The residue $\text{Res}(f, z_k)$ is the coefficient of $(z-z_k)^{-1}$ in the Laurent series expansion of $f(z)$ around the pole $z_k$
• Enables the evaluation of complex contour integrals by computing residues (e.g., using the Cauchy integral formula for derivatives)
• Widely used in applied mathematics, physics, and engineering (e.g., inverse Laplace transforms, Fourier analysis, quantum field theory)

Singularities and Poles

Types of Singularities

• A singularity of a complex function $f(z)$ is a point where $f(z)$ is not analytic (not differentiable)
• Removable singularities can be eliminated by redefining the function at the singular point (e.g., $\sin(z)/z$ at $z=0$)
• Essential singularities cannot be removed and cause the function to behave erratically near the singular point (e.g., $e^{1/z}$ at $z=0$)
• Branch points are singularities that arise from multi-valued functions (e.g., logarithms, fractional powers)

Poles and Their Classifications

• A pole is a type of isolated singularity where the function tends to infinity as $z$ approaches the singular point
• The order of a pole is the degree of the lowest-order term with a negative exponent in the Laurent series expansion around the pole
• Simple poles have order 1 and are characterized by a single term $(z-z_0)^{-1}$ in the Laurent series
• Higher-order poles (order $\geq 2$) involve multiple terms with negative exponents (e.g., $(z-z_0)^{-2}, (z-z_0)^{-3}$, etc.)
• The residue at a pole can be computed using the Cauchy integral formula for derivatives or by expanding the Laurent series