Impure Mathematics

Richard A. Gibbs

The Best Of The Journal of Irreproducible Results, 1983

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.