Courses / Mathematics / Complex analysis: holomorphic functions, contours and mappings Mathematics · unit version 1 Complex analysis: holomorphic functions, contours and mappings Proof-centered lessons on complex differentiability, contour theorems, analytic and Laurent series, residues, root counts, conformal maps and harmonic boundary values. Written arguments require mathematical review. The finite monodromy proof includes stable analytic-element chains and compact homotopy strips. General homological Cauchy remains a stated theorem dependency; the Riemann mapping theorem, Picard theorems and Riemann surfaces remain further study. Guided work is not independently assessed course equivalence.
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What is the real derivative of a map from R² to R²? Complex differentiability imposes additional structure on a real linear derivative.
Compare reasoning and find a refresher It is a real linear map represented by a 2×2 Jacobian, with a remainder small compared with the input displacement. Existence of coordinate partial derivatives alone does not establish this approximation.
Review differentiability as a linear approximation
Why can a vector field have zero curl locally but a nonzero integral around a closed curve? Complex contour theorems also require domain and hole hypotheses.
Compare reasoning and find a refresher A hole may prevent a closed curve from contracting within the domain. Local derivative conditions alone do not provide a global potential; the angular field around a missing origin is a standard counterexample.
Review holes and path independence
What does uniform convergence add to pointwise convergence on a contour or interval? Series, integration and analytic limits require justified interchange conditions.
Compare reasoning and find a refresher For each tolerance it supplies one index controlling every input in the domain. Pointwise convergence permits the index to depend on the input, so it need not control an integral or a moving point.
Review uniform convergence
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Complex analysis: domains, contours and proof obligations
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Lesson details and objectives Connect multiplication with rotation and scaling.
Connect multiplication with rotation and scaling. Prove modulus inequalities using conjugation. Describe disks and perpendicular bisectors algebraically. 6 practice tasks.
Lesson details and objectives Derive the exponential addition law from absolutely convergent series.
Derive the exponential addition law from absolutely convergent series. Find the periods and fibers of the exponential. Explain why complex sine and cosine are unbounded. 6 practice tasks.
Lesson details and objectives Specify a domain and argument range for a logarithm branch.
Specify a domain and argument range for a logarithm branch. Prove the derivative of a holomorphic logarithm is 1/z. Use winding to diagnose failure of a global logarithm. 6 practice tasks.
Lesson details and objectives Distinguish differentiability at a point from holomorphy on an open set.
Distinguish differentiability at a point from holomorphy on an open set. Derive the Cauchy–Riemann equations as a necessary condition. Use real differentiability to make those equations sufficient. 6 practice tasks.
Lesson details and objectives Compute contour integrals through piecewise smooth parametrizations.
Compute contour integrals through piecewise smooth parametrizations. Track orientation and reparametrization. Prove and apply the length-times-maximum estimate. 6 practice tasks.
Lesson details and objectives Prove endpoint evaluation for a holomorphic primitive.
Prove endpoint evaluation for a holomorphic primitive. Construct a primitive from path-independent integrals. Detect a global obstruction using a nonzero closed integral. 6 practice tasks.
Lesson details and objectives Prove triangular contour cancellation without assuming a continuous derivative.
Prove triangular contour cancellation without assuming a continuous derivative. Use domain hypotheses when extending local cancellation. Distinguish a singularity inside a contour from one outside the domain of integration. 6 practice tasks.
Lesson details and objectives Compute the integer winding number of oriented closed curves.
Compute the integer winding number of oriented closed curves. State Cauchy’s theorem in a form that respects holes. Track inner-boundary orientation in multiply connected regions. 6 practice tasks.
Lesson details and objectives Derive the interior value formula by subtracting the singular part.
Derive the interior value formula by subtracting the singular part. Compute derivative integrals with factorials and orientation. Explain why holomorphy forces derivatives of every order. 6 practice tasks.
Lesson details and objectives Bound derivatives from circle bounds using Cauchy’s formula.
Bound derivatives from circle bounds using Cauchy’s formula. Prove bounded entire functions are constant. Derive existence of polynomial roots without circular use of factorization. 6 practice tasks.
Lesson details and objectives Prove interior maximum rigidity using the Cauchy mean formula.
Prove interior maximum rigidity using the Cauchy mean formula. Explain the local multiplicity mechanism behind open mapping. Apply the maximum principle with the correct compact boundary hypotheses. 6 practice tasks.
Lesson details and objectives Derive Taylor coefficients from Cauchy’s formula.
Derive Taylor coefficients from Cauchy’s formula. Prove convergence on smaller disks with a geometric remainder bound. Distinguish analytic radius from a chosen boundary expansion point. 6 practice tasks.
Lesson details and objectives Factor a nonzero holomorphic germ by its zero multiplicity.
Factor a nonzero holomorphic germ by its zero multiplicity. Prove identity from interior accumulation of zeros. Distinguish boundary accumulation from interior uniqueness. 6 practice tasks.
Lesson details and objectives Derive positive and negative powers from two boundary kernels.
Derive positive and negative powers from two boundary kernels. Compute different expansions on different annuli. Recover Laurent coefficients by contour integration. 6 practice tasks.
Lesson details and objectives Classify isolated singularities by their Laurent principal parts.
Classify isolated singularities by their Laurent principal parts. Prove boundedness implies removability. Use reciprocal behavior to characterize poles and prove density near essential points. 6 practice tasks.
Lesson details and objectives Extract the coefficient that survives closed integration.
Extract the coefficient that survives closed integration. Compute simple and higher-order pole residues. Apply the residue theorem with winding and orientation. 6 practice tasks.
Lesson details and objectives Choose contours whose auxiliary pieces have justified limits.
Choose contours whose auxiliary pieces have justified limits. Evaluate rational real integrals from upper-half-plane poles. Distinguish ordinary improper convergence from principal value. 6 practice tasks.
Lesson details and objectives Relate logarithmic-derivative residues to zero and pole orders.
Relate logarithmic-derivative residues to zero and pole orders. Count zeros minus poles from a boundary image winding. Check boundary nonvanishing before interpreting argument change. 6 practice tasks.
Lesson details and objectives Prove root-count invariance using a boundary homotopy.
Prove root-count invariance using a boundary homotopy. Choose a dominant comparison term with a strict boundary inequality. Count zeros with multiplicity without locating each root. 6 practice tasks.
Lesson details and objectives Track poles and infinity in fractional-linear maps.
Track poles and infinity in fractional-linear maps. Prove invertibility from a nonzero determinant. Construct a half-plane-to-disk map and verify its image. 6 practice tasks.
Lesson details and objectives Prove angle preservation from a nonzero complex derivative.
Prove angle preservation from a nonzero complex derivative. Distinguish local conformality from global injectivity. Analyze powers, exponentials and disk automorphisms with explicit domains. 6 practice tasks.
Lesson details and objectives Derive harmonicity of real and imaginary parts of a holomorphic function.
Derive harmonicity of real and imaginary parts of a holomorphic function. Construct a conjugate from a closed differential on a simply connected domain. Use a punctured-plane example to separate local and global conjugates. 6 practice tasks.
Lesson details and objectives Derive the disk Poisson kernel and its normalization.
Derive the disk Poisson kernel and its normalization. Construct harmonic extensions of continuous boundary data. Prove boundary convergence and uniqueness using positivity and the maximum principle. 6 practice tasks.
Lesson details and objectives Continue a holomorphic germ using overlapping neighborhoods and identity uniqueness.
Continue a holomorphic germ using overlapping neighborhoods and identity uniqueness. Track logarithm and square-root changes around a loop. State the monodromy theorem with both existence and simple-connectivity hypotheses. 6 practice tasks.
Lesson details and objectives Construct segment and seam margins rather than assuming one global branch.
Define continuation by locally coherent germs and prove uniqueness. Build a finite adapted chain from compactness of the parameter interval. Prove uniform perturbation stability with fixed endpoints. 6 practice tasks.
Lesson details and objectives Build finite compatible strips, then prove the global monodromy conclusion and its limits.
Turn local path stability into finite homotopy transport. Prove local holomorphy of the path-independent endpoint assignment. Audit changed loops and separate sufficient hypotheses from one-loop behavior. 6 practice tasks.
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