(an old classic I ran across on USENET in 1983)
Once upon a time (1/T), pretty Polly Nomial was strolling
across a field of vectors when she came to the boundary of a
singularly large matrix. Now Polly was convergent, and her mother had
made it an absolute condition that she never enter such an array
without her brackets on. Polly, however, who had changed variables
that morning and was feeling particularly badly behaved, ignored this
condition and made her way in amongst the complex elements.
Rows and columns closed in on her from all sides. Tangents
approached her surface. She became tensor and tensor. Quite
suddenly, two branches of a hyperbola touched her at a single point.
She oscillated violently, lost all sense of directrix, and went
completely divergent. As she reached a turning point, she tripped
over a square root that was protruding from the Erf, and plunged
headlong down a steep gradient. When she rounded off once more, she
found herself inverted, apparently alone, in a non-Euclidean space.
She was being watched, however. That smooth operator, Curly Pi, was
lurking inner product. As his eyes devoured her curvilinear
coordinates, a singular expression crossed his face. He wondered, was
she convergent? He decided to integrate improperly at once.
Hearing a common fraction behind her, polly rotated and saw
Curly Pi approaching with his power series extrapolated. She could
see at once by his degenerative conic and his dissipative terms that
he was bent on no good.
"Arcsinh", she gasped.
"Ho, ho", he said. "What a symmetric little asymptote you
have. I can see that your angles have lots of secs."
"Oh, sir", she protested, "keep away from me. I haven't got
my brackets on."
"Calm yourself, my dear", said our suave operator. "Your
fears are purely imaginary."
I, I, she thought, perhaps he's not normal but homologous.
"What order are you?" the brute demanded.
"Seventeen", replied Polly.
Curly leered. "I suppose you've never been operated on?"
"Of course not", Polly replied quite properly, "I'm absolutely
convergent."
"Come, come", said Curly. "Let's off to a decimal place I
know and I'll take you to the limit."
"Never", gasped Polly.
"Abscissa", he swore, using the vilest oath he knew. His
patience was gone. Coshing her over the coefficient with a log until
she was powerless Curly removed her discontinuities. He stared at her
significant places and began smoothing her points of inflection.
Poor, poor Polly. The algorithmic method was now her only hope. She
felt his hand tending to her asymptotic limit. Her convergence would
soon be gone forever.
There was no mercy, for Curly was a Heavyside operator.
Curly's radius squared itself. Polly's loci quivered. He integrated
by parts. He integrated by partial fractions. After he cofactored,
he performed Runge-Kutta on her. The complex beast even went all the
way around and did a contour integration. What an indignity - to be
multiply connected on her first integration. Curly went on operating
until he had satisfied her hypothesis, then he exponentiated and
became completely orthogonal.
When Polly got home that night, her mother noticed that she
was no longer piece-wise continuous, but had been truncated in several
places and it was to late to differentiate now. As the months went
by, Polly's denominator increased monotonically. Finally, she went to
L'Hospital and generated a small but pathological function that left
surds all over the place and drove Polly to deviation.
The moral of our sad story is this:
"If you want to keep your expressions convergent, never allow them a
single degree of freedom."
-the end
-Aunty Derivative
