A Bootcamp to Complex Analysis

Explore disks and circles in the complex plane, define open disks as |z - z0| < r and circles as |z - z0| = r, using the distance formula.

Explore open and closed sets in the complex plane, define interior and boundary points, and learn how closure combines a set with its boundary, through disk and circle examples.

Define connectedness in the complex plane as not being splittable into two disjoint open sets; open sets let any two points connect by a successive line segment.

Explore bounded and unbounded sets in the complex plane using a disk of radius r centered at origin. See real and imaginary axes frame examples and infinity spanning directions.

Visualize plotting complex functions on the z and w planes using polar form, modulus and argument, exploring mappings like z^2 and z^2+1 with center shifts.

Explore complex sequences, finite and infinite, defined by explicit formulas or recursion, with complex numbers on the Argand plane and examples like the complex Fibonacci sequence.

Define the limit of complex sequences using the epsilon criterion and illustrate convergence as n tends to infinity with examples like 1/n, (a+ib)/n^p, and q^n.

Explore the rules for limits in complex analysis, including limit laws, convergence, and how sequences and complex numbers behave under limit rules, plus the squeeze theorem.

Explore how the exponential of a complex variable z = x + i y combines e^x with rotation e^{i y}, using Euler's formula cos y + i sin y.

Explore the exponential and gamma functions through area under the curve from zero to infinity, showing ∫_0^∞ x^n e^{-x} dx equals n!, and gamma(n) = ∫_0^∞ x^{n-1} e^{-x} dx.

Explore Euler's identity by deriving e^x as a Maclaurin series, substituting x = i theta, and obtaining e^{i theta} = cos theta + i sin theta.

Explore analytic functions as differentiable, single-valued complex functions on open sets. Polynomials are entire; rational functions are analytic where the denominator is nonzero; conjugate and real-part functions are not analytic.

Explore examples of analytic functions by testing limits along two paths; show that w = z z* is differentiable only at the origin, while w = z^2 is differentiable everywhere.

Explore the Cauchy-Riemann equations as conditions for differentiability of complex functions, using z^2 with u = x^2 − y^2 and v = 2xy, and relate f'(z) = 2z.

Derive Cauchy-Riemann equations in polar form by expressing x and y through r and theta, applying chain rule to u and v, and using z = r e^{i theta}.

explains what it means for the Cauchy-Riemann equations to be satisfied, showing the complex derivative exists and is independent of path as delta z tends to zero under CR conditions.

Explore how the exponential mapping e^z carries the z plane to the w plane, turning horizontal lines into rays and vertical lines into circles, with magnitude and phase.

Explore complex trigonometric functions through Euler’s formula, defining cos z and sin z as analytic and entire, and examine their addition formulas, period two pi, and derivatives.

Explore hyperbolic complex functions by deriving complex sine and cosine from real sine and cosine and hyperbolic sine and cosine for z = x + i y.

Explore analytic functions in open, connected domains, showing that zero derivative forces constancy; connect to Cauchy–Riemann, real-analytic intuition, and modulus considerations.