Complex Integration and Contour Integration

Overview

• Intuition of Integration
  • Real Integration: Area under a curve, mass of a rod.
  • Complex Integration: Sum of small components using complex increments.

Fundamental Concepts

• Real Integration: Integral as sum of rectangles, height is the function value, width is an increment in x (dx).
• Complex Function: Use f(z) as density, integrate using dz (complex increment).
• Contour Integration
  • Path must be specified when integrating in the complex plane.
  • Closed contours can use special integration symbols.

Calculating Complex Integrals

• Parametrization: Transform real path to complex path using a function v.
• Separate Real and Imaginary Parts: Integrate separately if possible.

Poar Vector Field

• Vector Field Construction: Attach vector to each point z, plot using conjugated vector values.
• Divergence and Curl:
  • Divergence is zero if satisfying Cauchy-Riemann equations.
  • Curl is zero if satisfying Cauchy-Riemann equations.

Work and Flux Interpretation

• Complex Number as Vector: Interpret f(z) and dz as vectors.
• Real Part of Integral: Represents work done by the field.
• Imaginary Part of Integral: Represents flux across the contour.

Cauchy's Theorem

• Holomorphic Functions: Integral around closed contour is zero if f is holomorphic inside the contour.
• Path Independence: Deforming the contour doesn't change the integral if function is holomorphic.

Important Integrals

• 1/z Integral: Integral around closed loop is 2πi.
• Higher Powers: Integral of 1/z^n for n>1 is zero.

Cauchy's Integral Formula

• Relates the value of a function inside a contour to its values on the contour.
• Can be differentiated to find derivatives of holomorphic functions.

Residue Theorem

• Residue Calculation: Extracting coefficient of 1/z in Laurent series.
• Integral Computation: Integral around closed loop equals sum of residues times 2πi.

Practical Applications

• Real Integral Calculation: Complex analysis can simplify calculation of difficult real integrals using contour methods.
• Example: Integral of 1/z^n for n>1 proves zero by symmetry of vector fields.

Additional Concepts

• Series Expansion: Useful for finding residues.
• Path Deformation: Use contour deformation to simplify integration.

Conclusion

• Complex Analysis: Provides powerful tools for evaluating integrals that are difficult to approach by real methods alone.
• Holomorphic Functions: Have a rigidity that allows unique determination from boundary values.