Hidden Singularity
Interactive 3D reproduction of Figure 2.14 from Tristan Needham's Visual Complex Analysis (Section 2.3.1, "The mystery of real power series"). The surface is the modular surface of the complex function h(z) = 1/(1 + z²) — the height at each point z = x + iy is |h(z)|. The two singularities at z = ±i erupt as "volcanoes" above the complex plane. Drag to rotate, scroll to zoom.
Height = |h(z)| = 1/|1 + z²|. The two peaks at z = ±i (the points (0, ±1) in the plane) shoot to infinity — the "volcanoes" of Figure 2.14. Drag to rotate, scroll to zoom.
The "deceptively tranquil" real function H(x) = 1/(1 + x²)
Restrict h(z) to the real axis and you get the smooth, bounded graph below — no hint of the singularities. The poles at ±i live off the real line, invisible to the real function alone. This is the mystery of Section 2.3.1: the radius of convergence of the real power series of H is governed by singularities that only the complex plane reveals.
Key Formulas (Section 2.3.1)
The real function \(H(x) = 1/(1 + x^2)\) is well-behaved for all real \(x\), yet its power series about \(x = k\) has radius of convergence \(R = \sqrt{1 + k^2}\) — a "strange formula" that cannot be explained using only real numbers.
The mystery unravels when we extend \(H\) to the complex function \(h(z) = 1/(1 + z^2)\), which agrees with \(H\) on the real axis. In the complex plane, \(h\) has two singularities, at \(z = i\) and \(z = -i\):
By Pythagoras, \(R = \sqrt{1 + k^2}\) is the distance from the centre \(k\) of the expansion to either of the two fixed points \(\pm i\) that lie off the real line, one unit from \(0\) in a direction at right angles to it:
The radius of convergence of a real power series is governed by singularities that may live off the real line, in the complex plane. The real function alone cannot explain its own radius of convergence — this is the book's recurring theme that the complex plane "explains" real analysis.