Singularity
Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.
The equation a sequence satisfies
Write the whole sequence as the coefficients of one series, and the recursion becomes an equation with a square in it. Solving the equation by the ordinary quadratic formula produces the closed form, the growth rate and the correction term, none of which the recursion offers.
The centre is a choice
A Taylor series is nearly always written about zero, and nothing about the construction prefers zero. Moving the centre moves the interval the series works on, and moving it repeatedly walks the function into places its first series could never reach.
A denominator that reaches past the radius
The Taylor series of ln(1 + x) is useless beyond x = 1 however many terms it is given. The same coefficients spent on a numerator and a denominator converge at x = 3, and at x = 100, because a polynomial cannot imitate a singularity and a quotient of two polynomials can.
Where two roots run into each other
Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.
An ellipse, not a disc
A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.
A slope for a curve that is no function
The folium x³ + y³ = 3xy loops back over itself, so no formula y = f(x) describes it, and yet at almost every point it has a perfectly good tangent. Differentiating the equation as it stands gives the slope, −(∂F/∂x) ÷ (∂F/∂y), and the only points where that fails are the ones where the curve turns vertical or crosses itself — which are exactly the points where it stops being a graph.
The tangent counts zigzags
Arrange the numbers 1 to n so that they go up, down, up, down. There are 1, 1, 2, 5, 16, 61, 272 such arrangements, and those are, multiplied by factorials, the Taylor coefficients of tan x and sec x — the odd lengths from the tangent, the even from the secant. Désiré André proved it in 1879 by splitting a zigzag at its largest number, and the split turns out to be the derivative of the tangent written out term by term.
Where to take the samples
Every numerical integral is a weighted sum of samples of the curve. Spacing the samples evenly gives errors that shrink like a power of their number; choosing where they go, as Gauss did in 1814, makes six samples exact for every polynomial up to degree eleven and turns the power into a geometric rate. The surprise is that the humble trapezoid rule, evenly spaced and exact only for straight lines, does the same thing on any smooth curve that goes once round a circle — and that a single corner defeats every one of them.
Named alongside it
The objects these essays reach for when they reach for this one.
Radius of convergenceApproximationConvergenceTaylor seriesAnalytic continuationAnalytic functionChebyshev polynomialComplex numbersFourier seriesOrthogonalityPolynomial approximationTangency