Wherein is related how that polygon of womanly virtue, young Polly Nomial
(our heroine) is accosted by that notorious villain ** Curly Pi **, and
factored (Oh horror!).
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 space.
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 degenerative 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 a lot 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. "Sevtenteen", 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 Pi. "Let's off to a decimal place 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 head with a log until she was powerless, Curly removed
her discontinuities. He stared at her significant places, and began smooting
out her points of inflection. 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-cutta on her.
The complex beast even went all the way around and did a contour integration.
Curly went on operating until he had satisfied her hypothesis, then
exponentiated and became completely orthogonal.
When Polly returned that night to her point of origin, her mother noticed
that she was no longer piecewise continuous, but had been truncated in several
places. 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 this sad story is this:
'If you want to keep your expressions convergent,
never allow them a single degree of freedom...'