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Impure Mathematics
by Richard A. Gibbs
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,
who had changed her variables that morning and was feeling particularly badly
behaved, 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 fraction behind her, Polly turned around 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
you're bubbling over with secs."
"O 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 homogenous then."
"What order are you," the brute demanded.
"Seventeenth," replied Polly.
Curly leered, "I suppose you've never been operated on yet?" he asked.
"Of course not," Polly cried indignantly. "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.
"Exchlf," 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
inflexion. Poor Polly... All was up. 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. He integrated by
parts. He integrated by partial fractions. The complex beast even went all
the way around and did a counter integration! What an indignity! To be
multiply connected ON HER FIRST INTEGRATION! Curly went on operating until he
was absolutely and completely orthogonal.
When Polly got home that evening, her mother noticed that she had been
truncated in several places.
But it was too late to differentiate now. 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 our sad story is this:
If you want to keep your expressions convergent, never allow them a
single degree of freedom.
(from the Best of JIR)
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