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.

Define a complex function as a rule mapping complex inputs to complex outputs, with domain and range. Show outputs as u(x,y)+i v(x,y) via z=x+iy, e.g., f(z)=z^2+1.

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 how iterating complex functions repeatedly applies a function to its own output, producing a sequence, with f^n as the nth iteration, not nth power, shown by z+1, 3z, z^2+2.

Explore Julia sets through the iteration of the quadratic polynomial f(z)=z^2+c, revealing chaotic and intricate behavior of complex numbers in the plane.

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 the limit of complex functions via epsilon-delta, illustrate with (z^2−1)/(z−1)→2 as z→1, and apply algebraic limit rules and boundedness near z0.

This lecture defines continuity for complex functions, stating that as z approaches z0, f(z) approaches f(z0) with f defined at z0 and the limit existing.

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 natural logarithm and the exponential function, including Euler's number e, Euler's identity, natural growth and continuous compounding, and the representation of complex numbers using exponential forms.

Examine natural and exponential growth rates using e^(rt) and continuous compounding, with r and t, and explore the derivative of the exponential function and the natural log (ln x).

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.

Examine the derivative of complex functions, highlighting analytic functions as an extension of real calculus and the two-dimensional nature of complex numbers.

Explore how the derivative of a complex function f(z) is defined by a path-independent limit, with rules for constants, sums, products, quotients, z^n, and non-differentiable cases real part and conjugate.

Demonstrate that differentiability at z0 implies continuity for a complex function, using epsilon-delta definitions in the complex plane. Also note that continuity does not necessarily imply differentiability in complex analysis.

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.

Examine how the Cauchy-Riemann equations relate the real and imaginary parts to differentiate complex functions, with examples like z^3, f(z)=conjugate z, and e^z to illustrate analyticity in the plane.

Represent the complex exponential e^z = e^x (cos y + i sin y), note its modulus |e^z| = e^x, its period 2π i, and apply e^{z+w} = e^z e^w.

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.