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
these conditions on the ground that they were unnecessary, and made her way
amongst the complex elements.
Rows and columns enveloped her on both 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 direction 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 alone, apparently in a
non-Euclidian space.
She was being watched however. That smooth operator, Curly Pi, was lurking
inner product. As his eyes devoured her
curvilinear co-ordinates, a singular expression crossed his face. Was she
still convergent, he wondered. He decided to integrate
at once.
Hearing a vulgar fraction behind her, Polly turned round and saw Curly Pi
approaching with his power series expanding. 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 absolutely bubbling over with 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 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 exclaimed indignantly. "I'm absolutely convergent".
"Come, come," said Curly, "let's go off to a decimal place, 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 Polly, all was up. She felt his digit tending to her
asymptotic limit. Her convergence was 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 - to be
multiply connected at 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 was 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 the story is this: If you want to keep your expressions
convergent, never allow them a single degree of freedom.
I first saw this story in 1973. It is no doubt older than that. I
would like to give credit to the author, but I don't know who it is.