^
Here's one ... Pardon the straight humor.
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.
