I M P U R E M A T H E M A T I C S
=====================================
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 sinse 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 noneuclidean 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 her 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 degenerate conic and his dissipative terms that he was
bent on no good.
"Arcsinh," she gasped.
"Ho, Ho," he said. "What a symmetric little asymtope 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 her only hope. She felt his
hand tending to her asymtotic limit. Her convergence wuold 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 rungekutta 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 orthagonal.
When Polly got home that night, her mother noticed that she was
no longer piecewise continuous, but had been truncated in several
places and it was too 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."
 Aunty Derivitave
