The Saga of Polly Nomial - Once upon a time, pretty Polly Nomial was skipping through a
field of vectors when she came to the edge of a singularly large matrix. Now Polly was
convergent, and her mother had made it an absolute condition that she never entered such an
array without her brackets on. But Polly had changed her variables that morning and had
been feeling particularly badly behaved, she ignored her mothers's condition on the grounds
that it was insufficient, and made her way in among the complex elements.
Rows and columns enveloped her on all sides. Tangents approached her surface. She grew
tensor and tensor. Quite suddenly, three branches of a hyperbola touched her at a single
point, she oscillated wildly and lost all sense of directrix. She tripped over a square root
protruding from the erf, and tumbled headlong down a steep gradient. When she was once
again in possesion of her variables, she found herself apparently 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 coordinates, a singular expression crossed his face.
Was she convergent? He wondered. He decided to integrate improperly at once. Hearing an
improper fraction behind her, Polly rotated and saw Curly approaching with his power series
extrapolated. She could tell at once from his degenerate conic and his dissipative terms that
he was bent to no good.
"Eureka!" she gasped.
"Ho, ho," said our operator. "What a symetric little asymptote you have. I bet your angles are
just dripping with secs."
"Stay away from me!" she said. "I haven't got my brackets on."
"Calm yourself, my dear," he said. "Your fears are purely imaginary."
"I, I," she thought, "Maybe he's not normal..Maybe he's even a homomorphism."
"What order are you?" the brute demanded.
"Seventeen," she replied.
Curly leered. "Enough of this idle chatter. Lets go to a decimal place I know, and I'll take you
to the limit."
"Never!" she gasped.
"Arcsinh!!!" He swore the vilest oath he knew. 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. She could feel his hand tending
towards her asymptotic limit. The algorithmic method was now her only hope. Her
convergence would soon be gone forever.
Curly's radius squared itself. Polly's loci quivered. He intergrated by parts. He intergrated by
partial fractions.The complex beast even went all the way around and did a contour
intergration. Curly went on operating until he was completely and totally exhausted of all his
primitive roots.
When Polly arrived home that night, her mother noticed that she had been truncated in
several places. But it was too late to differentiate now. Nine transformations later, she went to
L'Hopital and generated a small but pathological function which left zeros and residues all
over the place and drove poor Polly to deviation.
The moral of this story is: If you want to keep your expressions convergent, keep them well
differentiated from complex operators.