Title: Impure Mathematics
 
 IMPURE MATHEMATICS
 
 Wherein it is related how that polygon of womanly virtue, young
 Polly Nomial (our heroine), is accosted by the notorious
 villain, Curly Pi, and factored (oh, horror!!).
 
 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 and 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 singular 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 dissapative terms
 that he was bent on no good.
 
 "Arcsinh," she gasped.
 "Ho, ho," he said.  "What a symmetric little asymptote you have.
 I an see your angles have a lot 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 leared.  "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 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 to 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 by
 parts.  He integrated by partial fractions.  After he
 cofactored, he performed Runge-Cutta on her.  The complex beast
 even went all the way around and did a contour integration. 
 Curly went on operating until he had satisfied her hypothesis,
 then he exponentiated and became suddenly 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 is 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 expressions convergent, never allow
 them a single degree of freedom..."
 

Wil Stark, wstark04 (at) pobox _dot_com

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