The function f(x) = 1/(1 + x²) is smooth and well-defined for all real x, yet its Taylor series around x = 0 converges only for |x| < 1. What is the correct explanation?
AThe function grows too steeply beyond x = 1 for polynomial approximation to keep up
BThe Taylor series converges only where the function is increasing, and 1/(1+x²) decreases for x > 0
CIn the complex plane, f has singularities at z = ±i, which are distance 1 from the origin — those singularities govern the radius of convergence
DThe function's Taylor coefficients become large enough at n = 1 to cause divergence
This is the key insight of complex Taylor series: the radius of convergence equals the distance to the nearest singularity *in the complex plane*, not on the real line. Real analysis gives no clue why convergence stops at |x| = 1 — the function is perfectly smooth there. Complex analysis reveals the hidden obstruction: f(z) = 1/(1+z²) has poles at z = i and z = -i, each exactly distance 1 from the origin. The singularities are invisible on the real axis but they control the real Taylor series.
Question 2 Multiple Choice
Two holomorphic functions f and g agree on a small open disk D. What must be true on their shared domain?
Af and g agree only within D; outside D their values may diverge
Bf and g agree everywhere on their entire shared domain
Cf and g agree provided they share the same singularities
Df and g agree only if D contains a zero of f − g
This is the **identity theorem** for holomorphic functions. Because a holomorphic function is *equal* to its Taylor series on any disk (not merely approximated), the Taylor coefficients at any center point completely determine the function everywhere. Two functions agreeing on an open set must have identical Taylor coefficients there, so they are identical on their entire shared domain. There is no real-analysis analogue — two smooth functions can match on an interval while differing elsewhere, but holomorphic functions cannot.
Question 3 True / False
In complex analysis, a holomorphic function equals its Taylor series exactly within the disk of convergence — not approximately, as in real analysis.
TTrue
FFalse
Answer: True
This is the central distinction between real and complex power series. In real analysis, Taylor series are approximations — even smooth functions are only guaranteed to match their Taylor series in the limit, and the error can be nonzero at any finite order. In complex analysis, if f is holomorphic on |z − z₀| < R, then f(z) = Σ aₙ(z−z₀)ⁿ exactly, with zero error, everywhere on that disk. This follows from Cauchy's Integral Formula for derivatives and gives holomorphic functions their striking rigidity.
Question 4 True / False
The radius of convergence of the Taylor series of a complex function is determined by the behavior of the function near the real expansion point.
TTrue
FFalse
Answer: False
The radius of convergence is the distance from the expansion center to the nearest **singularity in the complex plane** — which may have no real manifestation at all. The function 1/(1+z²) is perfectly well-behaved on the real line everywhere, yet its Taylor series around z = 0 has radius of convergence 1 because its singularities z = ±i lie one unit away in the complex plane. Real-line behavior near the center is irrelevant; it is the global complex geometry (singularity locations) that governs convergence.
Question 5 Short Answer
Explain why the Taylor series of 1/(1 + z²) around z = 0 has radius of convergence exactly 1, even though the real function 1/(1+x²) is smooth and bounded for all real x.
Think about your answer, then reveal below.
Model answer: The radius of convergence equals the distance from the expansion center to the nearest singularity in the complex plane. The function 1/(1+z²) has singularities where 1+z² = 0, i.e., at z = i and z = -i. Both are exactly distance 1 from the origin. The series converges on the disk |z| < 1 — the largest disk centered at 0 that contains no singularity. On the real line, z = ±i are invisible, which is why the real function seems to have no obstruction to convergence beyond |x| = 1. Complex analysis reveals that the apparent mystery of real Taylor series is always resolved by the singularity structure in the complex plane.
This is historically one of the great clarifying results of complex analysis. Before the complex plane was understood, mathematicians were puzzled by power series that 'stopped working' at points where the real function was smooth. Cauchy and Riemann's insight was that the complex plane is the natural domain for analytic functions, and convergence is determined by the complex geometry — not the real geometry alone.