IMPURE MATHEMATICS

                         IMPURE MATHEMATICS

     Once upon a tim(l/t) pretty little Polly Nomial was
strolling across a field of vectors when she came across a
singularly large matrix.

     Now Polly was convergent  and her mother had made it an
absolute condition that she must never enter an array without her
brackets on.  Polly, however, who had changed her variables this
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 approached a turning point she
tripped over a square root that was protruding from the erf and
plunged headlong down a steep gradient.  When she was
differentiated once more she found herself completely 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 co-ordinates 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 are absolutely 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 smooth operator, "your
fears are purely imaginary."

     "i,i" she thought.  "Purhaps 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 purely
convergent."

     "Come come," said Curly.  "Lets off to a decimal place I
know and I will 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 all her discontinuities.  He
stared at her significant places and began smoothing her points
of inflection.  Poor Polly. All was up.  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.  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 on her
first integration.  Curly went on operating until he was
absolutely and completely orthagonal.

     When Polly got home that evening her mother noticed that she
had been truncated in several parts.  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 our sad story is this: if you want to keep your
expressions convergent, never allow them a single degree of
freedom.


     Reprinted from a 1967 edition of the California Engineer.