Impure Mathematics
-------------------
Wherein is related how that polygon of womanly virtue, young Polly
Nomial (our heroine) is accosted by that 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 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 degenerative 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 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. 'Sevtenteen',
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 Pi. 'Let's off to a decimal
place 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 head with a log until she
was powerless, Curly removed her discontinuities. He stared at
her significant places, and began smooting 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
exponentiated and became completely orthogonal.
When Polly returned that night to her point of origin,
her mother noticed that she was no longer piecewise continuous,
but had been truncated in several places. 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 this sad story is this:
'If you want to keep your expressions convergent,
never allow them a single degree of freedom...'
