IMPURE MATHEMATICS
 
   Wherein it  is related how that  paragon of womanly  virtue,  young
Polly Nomial  (our heroine),   is accosted  by the  notorious villian,
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
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 is was  insufficient,  and make her way in
among the complex elements.   Rows and  columns closed in on here 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 steed gradient.    When she rounded  off once
more, she found herself inverted, apparently aline, in a non-euclidian
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 an  saw Curly Pi
approaching with his power series extrapolated.. She could see at once
by his degenerate  conic an 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 angels have a lit of secs.'
   'Oh sir,' she  protested.   'Keep away form 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 your?' 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 go 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 know.   His patience
was gone.   Cohsing 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 heavysided 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-kutta on  here.   The complex beast even  went all the
way around  and did a contour  integration.   Curly went  on operation
until he  had satisfied her  hypothesis.   Then  he  exponentiated and
became completed orthogonal.
   When Polly got home that night,  her mother noticed that she was no
longer  piecewise continuous,    and  had  been truncated  in  several
places,  But is was too late to differentiate how.   A the months went
by, Polly's denominator increased monotonically,   Finally she went to
L'hospital and generated a small  but pathological function which left
surds all over the place an 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.'
 
 
 
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                 "To err is human -- to moo, bovine."
 
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