Impure Mathematix
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 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 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 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
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."