Complex differentiability imposes much stronger structure than real differentiability, which is why local hypotheses can lead to contour formulas and global restrictions. Lebl develops that chain through holomorphic functions, Cauchy's theorem, Laurent series and residues. Beck and collaborators offer a concise first-course presentation of holomorphic functions and contour methods. Use the second account to compare a difficult argument, while keeping one consistent set of conventions for contours and singularities.
Guide to Cultivating Complex Analysis: Working the Complex Field — Jiří Lebl. Read 2.1 on holomorphic functions, 3.2-3.3 on Cauchy's theorem and consequences, 4.4 on Laurent series, and 5.1-5.4 on zeros, singularities and residues. Review line integrals in 3.1 if needed.
Check complex differentiability from the definition before relying on the Cauchy-Riemann equations. Draw the contour and mark its interior singularities before selecting a theorem. Expand a function on two different annuli to see why the domain belongs to a Laurent series statement. In a residue calculation, separate the local coefficient calculation from the global contour argument. Record exactly where holomorphicity and the geometry of the domain enter each proof.
Connect local holomorphicity to series expansions and residue methods.
Read 2.1 on holomorphic functions, 3.2-3.3 on Cauchy's theorem and consequences, 4.4 on Laurent series, and 5.1-5.4 on zeros, singularities and residues. Review line integrals in 3.1 if needed.
Source · Checked 2026-10-16
Compare a first contour argument or singularity classification with a second exposition.
Find holomorphic functions and contour methods through the listed author edition; exact section numbering was not inherited.
Source · Checked 2026-10-11
Holomorphic-function and residue exercises. Exercises 2.1.1-2.1.6 for differentiability and Cauchy-Riemann; exercises embedded in sections 4.4 and 5.3 for Laurent series and residues.. Exercises provided; public solutions not verified.
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
Determine every point where f(z) = absolute value of z squared is complex differentiable, using the derivative definition as well as the Cauchy-Riemann equations.
Expand |z+h| squared. Compare increments along the real and imaginary axes, and check the origin for arbitrary complex increments.
For , , so the difference quotient is . Along real it tends to ; along purely imaginary it tends to . These two complex numbers are equal only at . At the quotient is , which tends to for arbitrary . Thus the derivative exists exactly at , where it equals .
For the Cauchy-Riemann check, and . The equations and give and . Their partial derivatives are continuous, so they also give sufficiency at the origin. Differentiability at this single point does not make holomorphic on any neighborhood.
If this was the sticking point:
Already comfortable with this? Skip the one-variable selection if you can do the first two checks and state the domain hypotheses of Cauchy's theorem. Take the several-variable route only after that; skip it if extension phenomena and positive-dimensional zero sets are already familiar.
Review the prerequisites · Compare the full subject collection
A first graduate course from holomorphic functions to conformal mapping and analytic continuation. Appendices review analysis prerequisites; exercises remain important for understanding.
A course-length tour of pseudoconvexity, CR geometry and the ∂̄-problem. Broad exposure rather than an exhaustive reference, with exercises used throughout.
A concise first course in holomorphic functions and contour methods. Work exercises before moving to several complex variables.
A first complex-analysis course with many exercises and geometric explanations. Suitable as a computational and conceptual complement to a proof-intensive text.
A first complex-variable course connecting contour methods, analytic continuation and classical applications.
Contour integration, Laurent expansions and residues lead to Fourier methods, conformal mapping and steepest-descent asymptotics.
Develops analytic methods in several complex variables. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.
Develops value-distribution techniques for meromorphic functions. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.
Relates complex geometry and analytic vanishing methods. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.
Develops classical complex analysis with a geometric perspective. Useful before studying Riemann surfaces or complex geometry.
A compact advanced undergraduate/graduate development with optional specialized material. A different pace from computational introductory texts.
Extends classical complex analysis toward elliptic functions, modular functions and analytic number theory. Sheaf-theoretic methods are not its focus.
A classical algebraic-function course at the interface with Riemann surfaces. Algebraic viewpoint and historical notation differ from a first contour-integral course.
A full complex-analysis course extending from holomorphic functions and residues to conformal geometry, elliptic functions and complex differential equations.
A course develops analytic functions, contour integration and residues before gamma and zeta functions, the prime number theorem and asymptotic analysis.
Several-variable power series and Hartogs phenomena lead to polynomial and holomorphic convexity, pseudoconvexity, the Levi problem, the d-bar equation and holomorphic mappings.
Analytic Lp spaces on the disc are developed through growth estimates, reproducing kernels, density and bounded Bergman projections.