
                        Impure Mathematix

     Wherein it is related how that polygon of womanly virtue, 
young Polly Nomial (our heroine), is accosted by that notorious 
villain Curly Pi, and factored (oh, horrors!).

     Once upon a time (1/t) pretty Polly Nomial was strolling 
across a field of vectors when she came to the boundary of a 
singularly large matrix.  Now Polly was convergent and her mother 
had made it an absolute condition that she 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 basis that it was 
insufficient, and made her way amongst the complex elements.  
Rows and columns closed in from all 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 directrix, and went 
completely divergent.  As she reached a turning point, she 
tripped over a square root that was protruding from the erf and 
plunged headlong down a steep gradient.  When she rounded off 
once more, she found herself inverted, apparently alone, in a non- 
euclidean space.

     She was being watched, however.  That smooth operator, Curly 
Pi, was lurking innerproduct.  As his eyes devoured her 
curvilinear coordinates, a singular expression crossed his face.  
He wondered, was she still convergent?  He decided to integrate 
improperly at once.

     Hearing a common fraction behind her, Polly rotated and saw 
Curly Pi approaching with his power series extrapolated.  She 
could see at once by his degenerate conic and dissipative terms 
that he was bent on no good.

     "Arcsinh!" she gasped.

     "Ho, ho," he said.  "What a symmetric little asymptote you 
have.  I can see your angles have lots of 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 not normal, but 
homologous."

     "What order are you??" the brute demanded.

     "Seventeen," replied Polly.

     Curly leered, "I suppose you've never been operated on."

     "Of course not," Polly replied quite properly, "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.

     "Abscissa!!!" he swore, using the vilest oath he knew.  His 
patience was gone.  Coshing her over the coefficient with a 
natural log until she was powerless, Curly removed her 
discontinuities.  He stared at her significant places, and began 
smoothing out her points of inflection.  Poor Polly.  The 
algorithmic method was now her only hope.  She felt his hand 
tending toward her asymptotic limit.  Her convergence would soon 
be gone forever.

     There was no mercy, for Curly was a heavyside operator.  
Curly's radius squared itself; Polly's loci quivered.  He 
integrated her by parts.  He integrated her by partial fractions. 
After he cofactored, he performed runge-cutta on her.  The 
complex beast even went all the way around and did a coutour 
integration.  Curly went on operating until he had satisfied her 
hypothesis.  Then, he exponentiated and became completely 
orthogonal.

     When Polly got home that night, her mother noticed that she 
was no longer piecewise continuous, but had been truncated in 
several places.  But, it was too late to differentiate now.  As 
the months went by, Polly's denominator increased monotonically.  
Finally, she went to l'hopital and generated a small but 
pathological function which left surds all over the place and 
drove Polly to deviation.

     The moral of our sad story is this:

     "If you want to keep your expression convergent, never allow 
them a single degree of freedom."

