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


"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


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)