Once upon a time (1/t), pretty little Polly Nomial was
strolling across a field of vectors, when she came to the edge of a
singularly large matrix. Now Polly was convergent, and her mother had
made it an absolute condition that she must never enter such an array
without her brackets on. Polly, however, had changed her variables that
morning, and, feeling particularly badly behaved, she ignored this condition
on the grounds that it was insufficient, and made her way in amongst
the complex elements.
Rows and columns enveloped her on all sides. Tangents approached her
surface. She became tensor and tensor. Quite suddenly, three 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 which was protruding
from the erf, and plunged headlong down a steep gradient. When she was
differentiated once more, she found herself, 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. Was she still convergent, he wondered.
He decided to integrate improperly at once.
Hearing a vulgar function behind her, Polly turned round, 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.
"Eureka" she gasped.
"Ho, ho!" he said. "What a symmetric little polynomial
you are. I can see that you are absolutely bubbling over with secs".
"Sir", she said, "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 homogeneous then?".
"What order are you?" the brute demanded.
"Seventeen", replied Polly.
Curly leered. "I suppose you've never been operated on yet?"
he said.
"Of course not" Polly cried indignantly. "I'm absolutely
convergent".
"Come, cone," said Curly. "Lets off to a decimal place
I know, and I'll take you to the limit".
"Never" gasped Polly.
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 to smooth her points of inflexion. Poor
Polly. All was lost. She felt his hand bonding to her asymptotic limit.
Her convergence would be gone for ever.
There was no mercy, for Curly was a heavyside operator. He integrated
by parts. He integrated by partial fractions. The complex beast even
went all the way round, and did a contour integration. What an indignity!
Curly went on operating until he was completely and absolutely orthogonal.
When Polly got hone that evening, her mother noticed that she had been
truncated in several places. But it was too late to differentiate now
--- the seeds having been sown. As the months went by, Polly increased
monotonically. Finally, she generated a small, but pathological, function,
which left surds all over the place, until she was driven to distraction.
The moral of this sad story is this: It you want to keep your expressions
convergent, never allow them a single degree of freedom.
