complex-fourier-analysis

name	complex-fourier-analysis
description	Solve and verify complex-variable and Fourier-analysis problems. Use for analytic functions, contour integrals, residues, branch cuts, Fourier series and transforms, convolution, spectral methods, sampling, and dispersion relations.
license	MIT

Complex and Fourier Analysis

Fix conventions and domains

State the complex domain, singularities, contour orientation, branch choices, transform normalization, frequency variable, and function space or decay assumptions. Mark endpoints and discontinuities. A branch-dependent answer without an explicit cut and argument convention is not complete.

Choose the method

• Establish analyticity before invoking Cauchy formulas or deforming a contour.
• Use residues after classifying poles and confirming arc or indentation contributions.
• Treat branch points and cuts directly; record values on both sides of the cut.
• For Fourier series, check periodic extension, symmetry, smoothness, and endpoint convergence.
• For transforms, verify integrability or use distributions explicitly; apply convolution and Parseval with matching conventions.
• For sampled data, account for windowing, leakage, resolution, aliasing, and normalization.

Verify

• Parameterize contours and check orientation, residues, and contributions at infinity.
• Differentiate or integrate the result when that recovers a simpler known expression.
• Confirm conjugate symmetry for real signals and compare time- and frequency-domain energy.
• Reconstruct the original signal and inspect error near discontinuities.
• Check limiting cases and numerical quadrature against the analytic result away from singularities.

Deliver

Report conventions, analytic assumptions, derivation, singularity and branch handling, result, convergence mode, and numerical or symbolic checks.

Source basis

Original synthesis informed by Lebl's openly licensed analysis and differential-equations texts recorded in ../../docs/TEXTBOOK_SOURCES.md.