Series, Complex Analysis and Operational Calculus (KPI, 2013)
A complete source map of 266 pages of KPI lecture notes and practicum spanning numerical and functional series, Fourier expansions, holomorphic functions, contour integration, Laurent series, residues, and Laplace-transform methods. The map seeds five GTK chapters whose conventions and formulas are independently checked.
This archive concatenates the 107-page lecture notes and 159-page practicum
compiled by Viktor O. Haidei, Lidiia B. Fedorova, Iryna V. Alieksieieva, and
Oleksandr O. Dykhovychnyi. Both original documents were read and visually
inspected page by page; their hashes are
0b1cf3faede328fc079839cafe7b66a484b0f76e6bf89f29879f4fb9c700e809
and 858b8496ca450929e064f97136ddcf6d4e23e628e2160b1e742794a9ef030188.
Coverage and routing #
The course first develops convergence of numerical, functional, power, and Taylor series, then Fourier series and spectra. It next introduces complex differentiability, the Cauchy–Riemann equations, contour integration, Laurent annuli, singularities, and residues. The final block treats Fourier integrals and the Laplace transform, including inversion, convolution, linear ODEs and systems, piecewise inputs, and Volterra equations.
GTK separates that material into five causal units: convergence and power series; orthogonal/Fourier expansion; holomorphicity; contour methods and residues; and Laplace transformation. This avoids hiding analytic hypotheses inside a broad “operational calculus” chapter.
Reliability boundary #
The source is useful for coverage but inconsistent in transform conventions and contains several consequential misprints. Logged defects include a missing term in a second-derivative Laplace formula, Fourier inversion and scaling errors, false global claims about the complex logarithm, sign or factor errors in Laurent and contour answers, phase-sign errors, time shifts used without the necessary Heaviside factor, transformed initial-condition errors, and double-counting in a Bessel-series calculation. GTK fixes conventions at the start of each chapter and verifies formulas against NIST DLMF and MIT course notes before using them.
Visual audit #
The lecture and practicum contain 69 grouped visual-ledger records, including periodic extensions and spectra, complex loci and contours, annuli, waveform gates, and transform sketches. None is copied. New local diagrams instead show orthogonal projection into Fourier modes, the local geometry of a holomorphic map, residue extraction on an annulus, and the time-domain/operator-domain structure of the Laplace transform.