Wherein it is related how that polygon of womanly virtue, young
Polly Nomial (our heroine) is accosted by that notorious villain
Curly Pi, and factored (oh, horrors!).
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 her variables that
morning and was feeling particularly badly behaved, ignored this
condition on the basis that it was insufficient, and made her way
amongst the complex elements. Rows and columns closed in 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
She was being watched, however. That smooth operator, Curly Pi,
was lurking innerproduct. As his eyes devoured her curvilinear
coordinates, a singular expression crossed his face. He wondered, was
she still 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 degenerate conic and 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 your angles have lots of Secs."
"Oh, Sir," she protested, "keep away from me. I haven't got my
"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."
"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 natural
log until she was powerless, Curly removed her discontinuities. He
stared at her significant places, and began smoothing out her points
of inflection. Poor Polly. The algorithmic method was now her only
hope. She felt his hand tending toward 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 her by
parts. He integrated her by partial fractions. After he cofactored,
he performed Runge-Cutta on her. The complex beast even went all the
way around and did a coutour 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 piecewise continuous, but had been truncated in several
places. But, it was too late to differentiate now. As the months
went by, Polly's denominator increased monotonically. Finally, she
went to L'Hopital and generated a small but pathological function
which 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 expression convergent,
never allow them a single degree of freedom."